Single Best Answer MCQs for Sub-Topics of Topic 8: Trigonometry

Question 1. In a right-angled triangle ABC, right-angled at B, the side opposite to angle A is the:

(A) Hypotenuse

(B) Side AB

(C) Side BC

(D) Side AC

Question 2. In a right triangle, the longest side is always the:

(A) Opposite side

(B) Adjacent side

(C) Hypotenuse

(D) Perpendicular

Question 3. If a right triangle has legs measuring $7\ \text{cm}$ and $24\ \text{cm}$, what is the length of the hypotenuse?

(A) $25\ \text{cm}$

(B) $31\ \text{cm}$

(C) $576\ \text{cm}$

(D) $625\ \text{cm}$

Question 4. In $\triangle PQR$, right-angled at Q, if PR = $13\ \text{units}$ and QR = $5\ \text{units}$, what is the length of PQ?

(A) $8\ \text{units}$

(B) $12\ \text{units}$

(C) $144\ \text{units}$

(D) $169\ \text{units}$

Question 5. The ratio of the opposite side to the hypotenuse in a right triangle defines which trigonometric ratio?

(A) Cosine

(B) Sine

(C) Tangent

(D) Secant

Question 6. If in a right triangle, the opposite side is $8$ and the adjacent side is $15$, what is the tangent of the angle?

(A) $\frac{8}{17}$

(B) $\frac{15}{17}$

(C) $\frac{8}{15}$

(D) $\frac{15}{8}$

Question 7. In $\triangle XYZ$, right-angled at Y, if XY = $9\ \text{cm}$ and YZ = $40\ \text{cm}$, what is $\sin Z$?

(A) $\frac{9}{41}$

(B) $\frac{40}{41}$

(C) $\frac{9}{40}$

(D) $\frac{40}{9}$

Question 8. If $\cos \theta = \frac{5}{13}$, what is $\tan \theta$?

(A) $\frac{12}{5}$

(B) $\frac{5}{12}$

(C) $\frac{12}{13}$

(D) $\frac{5}{13}$

Question 9. The reciprocal of $\sin \theta$ is:

(A) $\cos \theta$

(B) $\sec \theta$

(C) $\text{cosec} \theta$

(D) $\cot \theta$

Question 10. If $\cot A = \frac{3}{4}$, what is the value of $\sin A$?

(A) $\frac{3}{5}$

(B) $\frac{4}{5}$

(C) $\frac{3}{4}$

(D) $\frac{4}{3}$

Question 11. Which of the following is equal to $\frac{1}{\sec \theta}$?

(A) $\sin \theta$

(B) $\cos \theta$

(C) $\tan \theta$

(D) $\cot \theta$

Question 12. If $\text{cosec} \theta = \frac{17}{8}$, find the value of $\cos \theta$.

(A) $\frac{15}{17}$

(B) $\frac{8}{17}$

(C) $\frac{17}{15}$

(D) $\frac{8}{15}$

Question 13. The ratio of the adjacent side to the opposite side is:

(A) $\tan \theta$

(B) $\cot \theta$

(C) $\sec \theta$

(D) $\text{cosec} \theta$

Question 14. If $\tan \theta = \frac{1}{\sqrt{3}}$, which angle does $\theta$ represent in a standard right triangle context?

(A) $30^\circ$

(B) $45^\circ$

(C) $60^\circ$

(D) $0^\circ$

Question 15. In $\triangle ABC$, right-angled at B, if $\angle A = 30^\circ$ and AC = $10\ \text{cm}$, find the length of BC.

(A) $5\ \text{cm}$

(B) $5\sqrt{3}\ \text{cm}$

(C) $10\ \text{cm}$

(D) $\frac{10}{\sqrt{3}}\ \text{cm}$

Question 16. In a right triangle, if the two acute angles are $\alpha$ and $\beta$, then $\alpha + \beta$ is equal to:

(A) $45^\circ$

(B) $60^\circ$

(C) $90^\circ$

(D) $180^\circ$

Question 17. The relationship $\sin^2 \theta + \cos^2 \theta = 1$ is valid for:

(A) Only acute angles $\theta$

(B) Only angles in a right triangle

(C) All real values of $\theta$ for which sine and cosine are defined

(D) Only angles $0^\circ < \theta < 90^\circ$

Question 18. Which of the following is NOT a fundamental trigonometric identity?

(A) $1 + \tan^2 \theta = \sec^2 \theta$

(B) $\sin^2 \theta - \cos^2 \theta = 1$

(C) $\cot \theta = \frac{\cos \theta}{\sin \theta}$

(D) $\text{cosec} \theta = \frac{1}{\sin \theta}$

Question 19. Simplify $\sin \theta \cdot \cot \theta$.

(A) $\sin \theta$

(B) $\cos \theta$

(C) $\tan \theta$

(D) $\text{cosec} \theta$

Question 20. If $\sin \theta = \frac{1}{3}$, find the value of $9 \cos^2 \theta$.

(A) 1

(B) 8

(C) 9

(D) $\frac{8}{9}$

Question 21. If $\tan \theta = 2$, find $\sec^2 \theta$.

(A) 3

(B) 4

(C) 5

(D) $\sqrt{5}$

Question 22. Simplify $\frac{1 - \sin^2 \theta}{1 - \cos^2 \theta}$.

(A) $\tan^2 \theta$

(B) $\cot^2 \theta$

(C) $\sec^2 \theta$

(D) $\text{cosec}^2 \theta$

Question 23. In a right triangle, if $\angle A$ is acute, $\sin A$ is always less than or equal to:

(A) 0

(B) $\frac{1}{2}$

(C) $\frac{\sqrt{3}}{2}$

(D) 1

Question 24. If $\theta$ is an acute angle, $\sqrt{1+\tan^2 \theta}$ is equal to:

(A) $\sin \theta$

(B) $\cos \theta$

(C) $\tan \theta$

(D) $\sec \theta$

Question 25. Simplify $(\sec \theta - \tan \theta)(\sec \theta + \tan \theta)$.

(A) $\sec^2 \theta + \tan^2 \theta$

(B) 1

(C) 0

(D) $\sec^2 \theta - \tan^2 \theta$

Question 1. What is the exact value of $\sin 45^\circ + \cos 45^\circ$?

(A) 1

(B) $\sqrt{2}$

(C) 2

(D) $\frac{1}{\sqrt{2}}$

Question 2. Evaluate $2 \tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ$.

(A) 0

(B) 1

(C) 2

(D) $\frac{1}{2}$

Question 3. The value of $\sin 0^\circ + \tan 90^\circ$ is:

(A) 0

(B) 1

(C) undefined

(D) $\sqrt{3}$

Question 4. If $\sin \theta = \cos \theta$ for $0^\circ \leq \theta \leq 90^\circ$, what is the value of $\theta$?

(A) $0^\circ$

(B) $30^\circ$

(C) $45^\circ$

(D) $60^\circ$

Question 5. Evaluate $\frac{\tan 60^\circ}{\sin 30^\circ + \cos 30^\circ}$.

(A) $\frac{\sqrt{3}}{2}$

(B) $\sqrt{3}$

(C) 1

(D) $\frac{2}{\sqrt{3}}$

Question 6. If $2 \sin \theta = \sqrt{3}$, what is the value of $\theta$ for $0^\circ < \theta < 90^\circ$?

(A) $30^\circ$

(B) $45^\circ$

(C) $60^\circ$

(D) $90^\circ$

Question 7. The value of $\tan (90^\circ - A)$ is:

(A) $\tan A$

(B) $\cot A$

(C) $\sec A$

(D) $\text{cosec} A$

Question 8. Simplify $\sin 70^\circ / \cos 20^\circ$.

(A) 0

(B) 1

(C) $\tan 70^\circ$

(D) $\cot 20^\circ$

Question 9. If $\sin A = \cos B$, where A and B are acute angles, then $A+B$ is equal to:

(A) $45^\circ$

(B) $60^\circ$

(C) $90^\circ$

(D) $180^\circ$

Question 10. If $\tan 2A = \cot (A - 18^\circ)$, where 2A is an acute angle, find the value of A.

(A) $18^\circ$

(B) $36^\circ$

(C) $24^\circ$

(D) $27^\circ$

Question 11. Evaluate $\sin^2 20^\circ + \sin^2 70^\circ$.

(A) 0

(B) 1

(C) 2

(D) $\sin 90^\circ$

Question 12. If $\cos (90^\circ - A) = 1/2$, where A is acute, find the value of $\sin A$.

(A) $\frac{\sqrt{3}}{2}$

(B) $\frac{1}{2}$

(C) 1

(D) 0

Question 13. The value of $\cot 1^\circ \cot 2^\circ \cot 3^\circ ... \cot 89^\circ$ is:

(A) 0

(B) 1

(C) -1

(D) undefined

Question 14. If $\cos \theta = \sin 30^\circ$, then $\theta$ (acute) is:

(A) $30^\circ$

(B) $60^\circ$

(C) $45^\circ$

(D) $90^\circ$

Question 15. Evaluate $\frac{\cos 80^\circ}{\sin 10^\circ} + \cos 59^\circ \text{cosec } 31^\circ$.

(A) 0

(B) 1

(C) 2

(D) -1

Question 16. If $\sin (A+B) = 1$ and $\cos (A-B) = 1$, where $0^\circ \leq A+B \leq 90^\circ$ and $A>B$, find A and B.

(A) $A=45^\circ, B=45^\circ$

(B) $A=90^\circ, B=0^\circ$

(C) $A=60^\circ, B=30^\circ$

(D) $A=30^\circ, B=60^\circ$

Question 17. The value of $(1 + \tan \theta + \sec \theta)(1 + \cot \theta - \text{cosec } \theta)$ is:

(A) 0

(B) 1

(C) 2

(D) -1

Question 18. Simplify $\tan 1^\circ \tan 2^\circ \tan 3^\circ ... \tan 89^\circ$.

(A) 0

(B) 1

(C) -1

(D) undefined

Question 19. If $\sin 3A = \cos (A - 26^\circ)$, where 3A is an acute angle, find the value of A.

(A) $29^\circ$

(B) $30^\circ$

(C) $26^\circ$

(D) $28^\circ$

Question 20. The value of $\frac{\tan 65^\circ}{\cot 25^\circ}$ is:

(A) 0

(B) 1

(C) $\sqrt{3}$

(D) $\frac{1}{\sqrt{3}}$

Question 21. Evaluate $\cos 1^\circ \cos 2^\circ ... \cos 180^\circ$.

(A) 1

(B) 0

(C) -1

(D) $\frac{1}{2}$

Question 22. If $\text{cosec } A = \sec B$, where A and B are acute angles, then $A+B$ is:

(A) $45^\circ$

(B) $60^\circ$

(C) $90^\circ$

(D) $180^\circ$

Question 23. The value of $\sin^2 30^\circ + \cos^2 30^\circ$ is:

(A) 0

(B) 1

(C) 2

(D) $\frac{1}{4}$

Question 24. If $4 \tan \theta = 3$, evaluate $\frac{4 \sin \theta - \cos \theta}{4 \sin \theta + \cos \theta}$.

(A) $\frac{1}{2}$

(B) $\frac{1}{3}$

(C) $\frac{1}{4}$

(D) $\frac{2}{3}$

Question 25. The value of $\sin^2 5^\circ + \sin^2 10^\circ + ... + \sin^2 85^\circ + \sin^2 90^\circ$ is:

(A) 8

(B) 9

(C) $8.5$

(D) $9.5$

Question 1. Which identity is equivalent to $\sin^2 \theta = 1 - \cos^2 \theta$?

(A) $1 + \tan^2 \theta = \sec^2 \theta$

(B) $\sin^2 \theta + \cos^2 \theta = 1$

(C) $\text{cosec}^2 \theta - \cot^2 \theta = 1$

(D) $\sec^2 \theta - 1 = \tan^2 \theta$

Question 2. Simplify $\sec^2 A (1 - \sin^2 A)$.

(A) $\tan^2 A$

(B) $\cot^2 A$

(C) 1

(D) 0

Question 3. $\text{cosec } \theta - \cot \theta$ is equal to:

(A) $\frac{1 - \cos \theta}{\sin \theta}$

(B) $\frac{1 + \cos \theta}{\sin \theta}$

(C) $\frac{1 - \sin \theta}{\cos \theta}$

(D) $\frac{1 + \sin \theta}{\cos \theta}$

Question 4. Simplify $\tan A / \sin A$.

(A) $\sin A$

(B) $\cos A$

(C) $\sec A$

(D) $\text{cosec } A$

Question 5. $(1 + \tan^2 A) \cot A$ is equal to:

(A) $\sin A \cos A$

(B) $\sec A \text{cosec } A$

(C) $\tan A \cot A$

(D) $\sin A \text{cosec } A$

Question 6. If $\sin \theta + \cos \theta = \sqrt{2}$, what is the value of $\sin \theta \cos \theta$?

(A) 0

(B) 1

(C) $\frac{1}{2}$

(D) $\frac{\sqrt{2}}{2}$

Question 7. Which of the following is a correct identity?

(A) $\sin \theta \sec \theta = \cot \theta$

(B) $\cos \theta \text{cosec } \theta = \tan \theta$

(C) $\tan \theta \cot \theta = 1$

(D) $\sin \theta \cot \theta = \tan \theta$

Question 8. Simplify $\frac{1 + \tan^2 A}{1 + \cot^2 A}$.

(A) $\tan^2 A$

(B) $\cot^2 A$

(C) $\sec^2 A$

(D) $\text{cosec}^2 A$

Question 9. The expression $\sin^4 \theta + \cos^4 \theta + 2 \sin^2 \theta \cos^2 \theta$ simplifies to:

(A) 0

(B) 1

(C) $\sin^2 \theta$

(D) $\cos^2 \theta$

Question 10. $\text{cosec}^2 A - 1$ is equal to:

(A) $\tan^2 A$

(B) $\cot^2 A$

(C) $\sec^2 A$

(D) $\sin^2 A$

Question 11. If $\sec \theta + \tan \theta = p$, then $\sec \theta - \tan \theta$ is equal to:

(A) $p$

(B) $\frac{1}{p}$

(C) $1-p$

(D) $\sqrt{1-p^2}$

Question 12. $\sqrt{\frac{1+\sin A}{1-\sin A}}$ is equal to:

(A) $\sec A + \tan A$

(B) $\sec A - \tan A$

(C) $\text{cosec } A + \cot A$

(D) $\text{cosec } A - \cot A$

Question 13. Which expression simplifies to $\text{cosec } A$?

(A) $\cos A \tan A$

(B) $\frac{1}{\sin A}$

(C) $\frac{1}{\cos A}$

(D) $\sin A \cot A$

Question 14. $\sin^2 \theta - \cos^2 \theta$ can be written as:

(A) $2 \sin^2 \theta - 1$

(B) $1 - 2 \cos^2 \theta$

(C) $\cos^2 \theta - \sin^2 \theta$

(D) Both A and B

Question 15. The value of $\sin \theta (\text{cosec } \theta - \sin \theta)$ is:

(A) $\sin^2 \theta$

(B) $\cos^2 \theta$

(C) $\tan^2 \theta$

(D) 1

Question 16. Simplify $(\text{cosec } A - \sin A)(\sec A - \cos A)(\tan A + \cot A)$.

(A) 0

(B) 1

(C) -1

(D) 2

Question 17. If $\sin \theta = x$ and $\cos \theta = y$, then:

(A) $x^2 - y^2 = 1$

(B) $x^2 + y^2 = 1$

(C) $y^2 - x^2 = 1$

(D) $xy=1$

Question 18. The expression $\frac{\sin^2 \theta}{\cos^2 \theta} + \frac{\cos^2 \theta}{\sin^2 \theta}$ is equal to:

(A) $\sec^2 \theta \text{cosec}^2 \theta$

(B) $\tan^2 \theta + \cot^2 \theta$

(C) Both A and B

(D) 1

Question 19. Simplify $\frac{1}{1 + \sin \theta} + \frac{1}{1 - \sin \theta}$.

(A) $2 \sec^2 \theta$

(B) $2 \text{cosec}^2 \theta$

(C) $2 \tan^2 \theta$

(D) $2 \cot^2 \theta$

Question 20. Which of the following is an identity?

(A) $\sin \theta = \sqrt{1-\cos^2 \theta}$

(B) $\cos \theta = \sqrt{1-\sin^2 \theta}$

(C) $\tan \theta = \sqrt{\sec^2 \theta - 1}$

(D) $\text{cosec } \theta = \sqrt{1 + \cot^2 \theta}$

Question 21. If $\sec A + \tan A = \frac{3}{2}$, then $\sin A$ is equal to:

(A) $\frac{5}{13}$

(B) $\frac{12}{13}$

(C) $\frac{5}{3}$

(D) $\frac{3}{5}$

Question 1. The measure of a right angle in degrees is:

(A) $0^\circ$

(B) $90^\circ$

(C) $180^\circ$

(D) $360^\circ$

Question 2. How many minutes are there in one degree?

(A) 60

(B) 100

(C) 3600

(D) 10

Question 3. The relationship between degree measure and radian measure is:

(A) $1^\circ = (\pi/180)$ radians

(B) $1 \text{ radian} = (180/\pi)^\circ$

(C) $1^\circ = (180/\pi)$ radians

(D) $1 \text{ radian} = (\pi/360)^\circ$

Question 4. Convert $45^\circ$ into radians.

(A) $\pi$ radians

(B) $\frac{\pi}{2}$ radians

(C) $\frac{\pi}{3}$ radians

(D) $\frac{\pi}{4}$ radians

Question 5. Convert $\frac{5\pi}{6}$ radians into degrees.

(A) $150^\circ$

(B) $120^\circ$

(C) $210^\circ$

(D) $300^\circ$

Question 6. An arc of a circle with radius $10\ \text{cm}$ subtends an angle of $2$ radians at the centre. What is the length of the arc?

(A) $5\ \text{cm}$

(B) $10\ \text{cm}$

(C) $20\ \text{cm}$

(D) $100\ \text{cm}$

Question 7. The area of a sector of a circle with radius $4\ \text{cm}$ and central angle $3$ radians is:

(A) $12\ \text{cm}^2$

(B) $18\ \text{cm}^2$

(C) $24\ \text{cm}^2$

(D) $48\ \text{cm}^2$

Question 8. A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?

(A) $6\pi$ radians

(B) $12\pi$ radians

(C) $360\pi$ radians

(D) $720\pi$ radians

Question 9. Find the degree measure of the angle subtended at the centre of a circle of radius $100\ \text{cm}$ by an arc of length $22\ \text{cm}$. (Use $\pi = \frac{22}{7}$)

(A) $12.6^\circ$

(B) $14^\circ$

(C) $12^\circ 36'$

(D) $14^\circ 36'$

Question 10. What is the angle in radians subtended by an arc of length $5\ \text{cm}$ at the centre of a circle with radius $5\ \text{cm}$?

(A) $\pi$ radians

(B) 1 radian

(C) 5 radians

(D) $\frac{1}{5}$ radian

Question 11. Convert $330^\circ$ into radians.

(A) $\frac{11\pi}{6}$ radians

(B) $\frac{5\pi}{3}$ radians

(C) $\frac{7\pi}{6}$ radians

(D) $\frac{11\pi}{12}$ radians

Question 12. Convert $\frac{7\pi}{4}$ radians into degrees.

(A) $280^\circ$

(B) $315^\circ$

(C) $300^\circ$

(D) $330^\circ$

Question 13. The ratio of the circumference of a circle to its diameter is:

(A) 2

(B) $\pi$

(C) $\pi/2$

(D) $2\pi$

Question 14. An angle of $1^\circ$ is equal to approximately how many radians?

(A) 0.01745 radians

(B) 0.01457 radians

(C) 1.745 radians

(D) 1.457 radians

Question 15. Two circles have arcs of the same length, subtending angles $60^\circ$ and $75^\circ$ at the centre. What is the ratio of their radii?

(A) $4:5$

(B) $5:4$

(C) $6:5$

(D) $5:6$

Question 16. The minute hand of a watch is $1.5\ \text{cm}$ long. How far does its tip move in 40 minutes? (Use $\pi = 3.14$)

(A) $3.14\ \text{cm}$

(B) $6.28\ \text{cm}$

(C) $9.42\ \text{cm}$

(D) $12.56\ \text{cm}$

Question 17. Find the radius of the circle in which a central angle of $60^\circ$ intercepts an arc of length $37.4\ \text{cm}$. (Use $\pi = \frac{22}{7}$)

(A) $35\ \text{cm}$

(B) $36\ \text{cm}$

(C) $37\ \text{cm}$

(D) $38\ \text{cm}$

Question 18. The angle in degrees of a sector with area $\frac{1}{4}$ of a circle with radius r is:

(A) $45^\circ$

(B) $60^\circ$

(C) $90^\circ$

(D) $180^\circ$

Question 19. Convert $\frac{\pi}{12}$ radians to degrees.

(A) $10^\circ$

(B) $12^\circ$

(C) $15^\circ$

(D) $18^\circ$

Question 20. What is the central angle in radians subtended by an arc of length $\pi\ \text{cm}$ in a circle of radius $18\ \text{cm}$?

(A) $\frac{\pi}{18}$ radians

(B) $\frac{1}{18}$ radian

(C) $\frac{\pi}{9}$ radians

(D) $\frac{1}{9}$ radian

Question 21. Find the angle in radians between the minute hand and hour hand of a clock at 6:00 PM.

(A) $\pi/2$ radians

(B) $\pi$ radians

(C) $3\pi/2$ radians

(D) $2\pi$ radians

Question 22. How many revolutions per minute does a wheel make if it turns through 450 radians in 1 minute?

(A) $\frac{225}{\pi}$ rpm

(B) $\frac{450}{\pi}$ rpm

(C) $\frac{225}{2\pi}$ rpm

(D) $\frac{450}{2\pi}$ rpm

Question 1. Using the unit circle, the coordinates of the point corresponding to the angle $90^\circ$ ($\frac{\pi}{2}$ radians) are:

(A) $(1, 0)$

(B) $(0, 1)$

(C) $(-1, 0)$

(D) $(0, -1)$

Question 2. The value of $\sin 270^\circ$ is:

(A) 1

(B) 0

(C) -1

(D) undefined

Question 3. The domain of $\sin x$ is:

(A) $\mathbb{R}$

(B) $[-1, 1]$

(C) $(-\infty, \infty)$

(D) All real numbers except $n\pi$ for integer n

Question 4. The range of $\cos x$ is:

(A) $\mathbb{R}$

(B) $[-1, 1]$

(C) $(-\infty, -1] \cup [1, \infty)$

(D) $(0, 2\pi]$

Question 5. In which quadrant is $\cos \theta < 0$ and $\tan \theta > 0$?

(A) Quadrant I

(B) Quadrant II

(C) Quadrant III

(D) Quadrant IV

Question 6. If the terminal side of an angle $\theta$ passes through the point $(-3, 4)$, what is the value of $\sin \theta$?

(A) $\frac{4}{5}$

(B) $-\frac{3}{5}$

(C) $\frac{3}{5}$

(D) $-\frac{4}{5}$

Question 7. The period of $\cos x$ is:

(A) $\pi$

(B) $2\pi$

(C) $\frac{\pi}{2}$

(D) $3\pi$

Question 8. The range of $\text{cosec } x$ is:

(A) $[-1, 1]$

(B) $\mathbb{R} - (-1, 1)$

(C) $(-\infty, -1] \cup [1, \infty)$

(D) Both B and C

Question 9. In which quadrant are all trigonometric functions positive?

(A) Quadrant I

(B) Quadrant II

(C) Quadrant III

(D) Quadrant IV

Question 10. The period of $\tan (2x)$ is:

(A) $\pi$

(B) $2\pi$

(C) $\frac{\pi}{2}$

(D) $4\pi$

Question 11. What is the value of $\cos (\frac{3\pi}{2})$?

(A) 1

(B) 0

(C) -1

(D) undefined

Question 12. The domain of $\tan x$ is all real numbers except:

(A) $n\pi, n \in \mathbb{Z}$

(B) $(2n+1)\frac{\pi}{2}, n \in \mathbb{Z}$

(C) $2n\pi, n \in \mathbb{Z}$

(D) $n\frac{\pi}{2}, n \in \mathbb{Z}$

Question 13. If $\sin \theta = -\frac{1}{2}$ and $\tan \theta = \frac{1}{\sqrt{3}}$, in which quadrant does $\theta$ lie?

(A) Quadrant I

(B) Quadrant II

(C) Quadrant III

(D) Quadrant IV

Question 14. The smallest positive period of $\sec x$ is:

(A) $\pi/2$

(B) $\pi$

(C) $2\pi$

(D) $4\pi$

Question 15. What is the value of $\tan (-\frac{\pi}{4})$?

(A) 1

(B) -1

(C) $\frac{1}{\sqrt{3}}$

(D) $-\sqrt{3}$

Question 16. If $\cos \theta = \frac{3}{5}$ and $\theta$ is in the fourth quadrant, find $\sin \theta$.

(A) $\frac{4}{5}$

(B) $-\frac{4}{5}$

(C) $\frac{3}{4}$

(D) $-\frac{3}{4}$

Question 17. The period of $y = \sin(3x)$ is:

(A) $2\pi$

(B) $\frac{2\pi}{3}$

(C) $3\pi$

(D) $\frac{\pi}{3}$

Question 18. The range of $\cot x$ is:

(A) $[-1, 1]$

(B) $\mathbb{R}$

(C) $(-\infty, -1] \cup [1, \infty)$

(D) $(0, \pi)$

Question 19. If the terminal side of an angle $\theta$ lies on the negative y-axis, then $\sin \theta$ is:

(A) 1

(B) 0

(C) -1

(D) undefined

Question 20. The period of $y = \cos(\frac{x}{2})$ is:

(A) $\pi$

(B) $2\pi$

(C) $4\pi$

(D) $\pi/2$

Question 21. In which quadrant is $\text{cosec } \theta > 0$ and $\cot \theta < 0$?

(A) Quadrant I

(B) Quadrant II

(C) Quadrant III

(D) Quadrant IV

Question 22. The value of $\sec (180^\circ)$ is:

(A) 1

(B) -1

(C) 0

(D) undefined

Question 1. The graph of $y = \cos x$ passes through which point?

(A) $(0, 0)$

(B) $(\frac{\pi}{2}, 1)$

(C) $(\pi, -1)$

(D) $(2\pi, 0)$

Question 2. The minimum value of the function $y = \sin x$ is:

(A) 0

(B) 1

(C) -1

(D) $-\infty$

Question 3. The graph of $y = \tan x$ has roots (x-intercepts) at values of $x$ where:

(A) $\sin x = 0$

(B) $\cos x = 0$

(C) $\tan x$ is undefined

(D) $\sec x = 0$

Question 4. Which trigonometric function has vertical asymptotes at $x = (2n+1)\frac{\pi}{2}$, where $n$ is an integer?

(A) $y = \sin x$

(B) $y = \cos x$

(C) $y = \tan x$

(D) Both $y = \tan x$ and $y = \sec x$

Question 5. The amplitude of the function $y = 5 \cos x$ is:

(A) 1

(B) 5

(C) $2\pi/5$

(D) $2\pi$

Question 6. The graph of $y = \sec x$ has the same vertical asymptotes as the graph of:

(A) $y = \sin x$

(B) $y = \cos x$

(C) $y = \tan x$

(D) $y = \cot x$

Question 7. What is the period of the function $y = \sin(x/2)$?

(A) $\pi$

(B) $2\pi$

(C) $4\pi$

(D) $\pi/2$

Question 8. The graph of $y = \cot x$ has roots (x-intercepts) at values of $x$ where:

(A) $\sin x = 0$

(B) $\cos x = 0$

(C) $\cot x$ is undefined

(D) $\tan x = 0$

Question 9. What is the equation of a vertical asymptote for the graph of $y = \tan x$?

(A) $x = 0$

(B) $x = \pi$

(C) $x = \pi/2$

(D) $y = 0$

Question 10. The graph of $y = \sin x$ is symmetric about the:

(A) x-axis

(B) y-axis

(C) Origin

(D) Line $y=x$

Question 11. Which function is an odd function?

(A) $\cos x$

(B) $\sec x$

(C) $\sin x$

(D) $| \cos x |$

Question 12. The graph of $y = \cos x$ is symmetric about the:

(A) x-axis

(B) y-axis

(C) Origin

(D) Line $y=x$

Question 13. Which function is an even function?

(A) $\sin x$

(B) $\tan x$

(C) $\cot x$

(D) $\cos x$

Question 14. The graph of $y = 2 \sin x$ has an amplitude of:

(A) 1

(B) 2

(C) $\pi$

(D) $2\pi$

Question 15. The period of $y = \tan(x/3)$ is:

(A) $\pi/3$

(B) $\pi$

(C) $3\pi$

(D) $6\pi$

Question 16. The graph of $y = \sin x$ reaches its maximum value at $x = \pi/2 + 2n\pi$. At which points does $y = \cos x$ reach its maximum value?

(A) $x = 0 + 2n\pi$

(B) $x = \pi + 2n\pi$

(C) $x = \pi/2 + 2n\pi$

(D) $x = 3\pi/2 + 2n\pi$

Question 17. The graph of $y = \cos x$ has roots (x-intercepts) at values of $x$ where:

(A) $x = n\pi$

(B) $x = 2n\pi$

(C) $x = (2n+1)\frac{\pi}{2}$

(D) $x = n\frac{\pi}{2}$

Question 18. The range of $y = 3 \sin x$ is:

(A) $[-1, 1]$

(B) $[-3, 3]$

(C) $[0, 3]$

(D) $(-\infty, \infty)$

Question 19. Which function has a period of $\pi$?

(A) $\sin x$

(B) $\cos x$

(C) $\tan x$

(D) $\sec x$

Question 20. The graph of $y = \text{cosec } x$ has vertical asymptotes at values of $x$ where:

(A) $\sin x = 0$

(B) $\cos x = 0$

(C) $\tan x = 0$

(D) $\cot x = 0$

Question 21. What is the period of the function $y = \cos(4x)$?

(A) $2\pi$

(B) $\pi$

(C) $\pi/2$

(D) $4\pi$

Question 22. The graph of $y = | \sin x |$ has a period of:

(A) $\pi/2$

(B) $\pi$

(C) $2\pi$

(D) $4\pi$

Question 23. The range of $y = 1 + \sin x$ is:

(A) $[0, 2]$

(B) $[-1, 1]$

(C) $[1, 2]$

(D) $[0, 1]$

Question 1. The formula for $\cos(A+B)$ is:

(A) $\cos A \cos B + \sin A \sin B$

(B) $\cos A \cos B - \sin A \sin B$

(C) $\sin A \cos B + \cos A \sin B$

(D) $\sin A \cos B - \cos A \sin B$

Question 2. $\tan(A-B)$ is equal to:

(A) $\frac{\tan A + \tan B}{1 - \tan A \tan B}$

(B) $\frac{\tan A - \tan B}{1 + \tan A \tan B}$

(C) $\frac{\tan A - \tan B}{1 - \tan A \tan B}$

(D) $\frac{\tan A + \tan B}{1 + \tan A \tan B}$

Question 3. $\cos 2A$ can be expressed as:

(A) $2 \sin^2 A - 1$

(B) $\sin^2 A - \cos^2 A$

(C) $1 - 2 \cos^2 A$

(D) $\cos^2 A - \sin^2 A$

Question 4. The value of $\sin 75^\circ$ is:

(A) $\frac{\sqrt{3}+1}{2\sqrt{2}}$

(B) $\frac{\sqrt{3}-1}{2\sqrt{2}}$

(C) $\frac{1+\sqrt{2}}{2}$

(D) $\frac{\sqrt{3}}{2}$

Question 5. $\tan 2A$ is equal to:

(A) $\frac{2 \tan A}{1 - \tan^2 A}$

(B) $\frac{2 \tan A}{1 + \tan^2 A}$

(C) $\frac{1 - \tan^2 A}{2 \tan A}$

(D) $\frac{1 + \tan^2 A}{2 \tan A}$

Question 6. The value of $\cos 105^\circ$ is:

(A) $\frac{\sqrt{3}+1}{2\sqrt{2}}$

(B) $\frac{\sqrt{3}-1}{2\sqrt{2}}$

(C) $-\frac{\sqrt{3}+1}{2\sqrt{2}}$

(D) $-\frac{\sqrt{3}-1}{2\sqrt{2}}$

Question 7. If $\tan A = \frac{1}{7}$ and $\tan B = \frac{1}{3}$, find $\tan(A+B)$.

(A) $\frac{1}{2}$

(B) $\frac{1}{3}$

(C) $1$

(D) $2$

Question 8. $\sin 3A$ is equal to:

(A) $3 \sin A - 4 \sin^3 A$

(B) $4 \sin^3 A - 3 \sin A$

(C) $3 \cos A - 4 \cos^3 A$

(D) $4 \cos^3 A - 3 \cos A$

Question 9. $1 + \cos 2\theta$ is equal to:

(A) $2 \sin^2 \theta$

(B) $2 \cos^2 \theta$

(C) $\sin^2 \theta$

(D) $\cos^2 \theta$

Question 10. The value of $\sin 22.5^\circ$ is:

(A) $\sqrt{\frac{1 - \cos 45^\circ}{2}}$

(B) $\sqrt{\frac{1 + \cos 45^\circ}{2}}$

(C) $\sqrt{\frac{1 - \sin 45^\circ}{2}}$

(D) $\sqrt{\frac{1 + \sin 45^\circ}{2}}$

Question 11. $\frac{1 - \tan^2 A}{1 + \tan^2 A}$ is equal to:

(A) $\sin 2A$

(B) $\cos 2A$

(C) $\tan 2A$

(D) $\cot 2A$

Question 12. $\cos 3A$ is equal to:

(A) $3 \sin A - 4 \sin^3 A$

(B) $4 \sin^3 A - 3 \sin A$

(C) $3 \cos A - 4 \cos^3 A$

(D) $4 \cos^3 A - 3 \cos A$

Question 13. The value of $\tan \frac{\theta}{2}$ can be expressed as:

(A) $\sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}}$

(B) $\sqrt{\frac{1 + \cos \theta}{1 - \cos \theta}}$

(C) $\frac{1 - \cos \theta}{\sin \theta}$

(D) Both A and C

Question 14. If $\sin A = \frac{3}{5}$, and A is acute, find $\sin 2A$.

(A) $\frac{24}{25}$

(B) $\frac{7}{25}$

(C) $\frac{6}{5}$

(D) $\frac{3}{10}$

Question 15. Simplify $\frac{\sin 2\theta}{1 + \cos 2\theta}$.

(A) $\tan \theta$

(B) $\cot \theta$

(C) $\tan 2\theta$

(D) $\cot 2\theta$

Question 16. The value of $\sin 15^\circ$ is:

(A) $\frac{\sqrt{3}+1}{2\sqrt{2}}$

(B) $\frac{\sqrt{3}-1}{2\sqrt{2}}$

(C) $\frac{1+\sqrt{2}}{2}$

(D) $\frac{\sqrt{3}}{2}$

Question 17. $\sin(A-B)$ is equal to:

(A) $\sin A \cos B + \cos A \sin B$

(B) $\sin A \cos B - \cos A \sin B$

(C) $\cos A \cos B + \sin A \sin B$

(D) $\cos A \cos B - \sin A \sin B$

Question 18. If $\cos A = \frac{4}{5}$ and A is acute, find $\cos 2A$.

(A) $\frac{7}{25}$

(B) $\frac{24}{25}$

(C) $\frac{8}{5}$

(D) $\frac{4}{10}$

Question 19. The value of $\cos^2 15^\circ - \sin^2 15^\circ$ is:

(A) $\frac{1}{2}$

(B) $\frac{\sqrt{3}}{2}$

(C) 1

(D) 0

Question 20. Simplify $\sin(60^\circ - \theta) + \sin(60^\circ + \theta)$.

(A) $\sin \theta$

(B) $\cos \theta$

(C) $\sqrt{3} \cos \theta$

(D) $\sqrt{3} \sin \theta$

Question 21. If $\tan A = \frac{a}{b}$, then $\frac{a \sin 2A + b \cos 2A}{a \sin 2A - b \cos 2A}$ is equal to:

(A) $\frac{a^2+b^2}{a^2-b^2}$

(B) $\frac{a^2-b^2}{a^2+b^2}$

(C) 1

(D) $\frac{2ab}{a^2+b^2}$

Question 22. The value of $\frac{\sin 2A}{1 - \cos 2A}$ is:

(A) $\tan A$

(B) $\cot A$

(C) $\sin A$

(D) $\cos A$

Question 23. $1 - \tan^2 A$ is equal to:

(A) $\sec^2 A$

(B) $\frac{\cos 2A}{\cos^2 A}$

(C) $\frac{\sec A}{\tan A}$

(D) $\text{cosec}^2 A$

Question 1. $2 \cos A \sin B$ is equal to:

(A) $\sin(A+B) + \sin(A-B)$

(B) $\sin(A+B) - \sin(A-B)$

(C) $\cos(A+B) + \cos(A-B)$

(D) $\sin(A-B) - \sin(A+B)$

Question 2. $\sin C - \sin D$ is equal to:

(A) $2 \cos \left(\frac{C+D}{2}\right) \sin \left(\frac{C-D}{2}\right)$

(B) $2 \sin \left(\frac{C+D}{2}\right) \cos \left(\frac{C-D}{2}\right)$

(C) $2 \cos \left(\frac{C+D}{2}\right) \cos \left(\frac{C-D}{2}\right)$

(D) $2 \sin \left(\frac{C+D}{2}\right) \sin \left(\frac{C-D}{2}\right)$

Question 3. Express $\cos 4x + \cos 2x$ as a product.

(A) $2 \sin 3x \sin x$

(B) $2 \cos 3x \cos x$

(C) $2 \sin 3x \cos x$

(D) $2 \cos 3x \sin x$

Question 4. Simplify $2 \sin 5\theta \sin 3\theta$.

(A) $\cos 8\theta + \cos 2\theta$

(B) $\cos 2\theta - \cos 8\theta$

(C) $\sin 8\theta + \sin 2\theta$

(D) $\sin 8\theta - \sin 2\theta$

Question 5. $\frac{\sin 5x + \sin 3x}{\cos 5x + \cos 3x}$ is equal to:

(A) $\tan 4x$

(B) $\cot 4x$

(C) $\tan x$

(D) $\cot x$

Question 6. The value of $\sin 105^\circ + \cos 105^\circ$ is:

(A) $\frac{1}{2\sqrt{2}}$

(B) $\frac{1}{\sqrt{2}}$

(C) $\sqrt{2}$

(D) 1

Question 7. Simplify $\frac{\sin x - \sin y}{\cos x + \cos y}$.

(A) $\tan \left(\frac{x+y}{2}\right)$

(B) $\cot \left(\frac{x+y}{2}\right)$

(C) $\tan \left(\frac{x-y}{2}\right)$

(D) $\cot \left(\frac{x-y}{2}\right)$

Question 8. Express $\cos A \cos B$ in terms of sum/difference of sines or cosines.

(A) $\frac{1}{2} [\sin(A+B) + \sin(A-B)]$

(B) $\frac{1}{2} [\sin(A+B) - \sin(A-B)]$

(C) $\frac{1}{2} [\cos(A+B) + \cos(A-B)]$

(D) $\frac{1}{2} [\cos(A-B) - \cos(A+B)]$

Question 9. Simplify $\frac{\sin 7\theta - \sin 5\theta}{\cos 7\theta + \cos 5\theta}$.

(A) $\tan \theta$

(B) $\cot \theta$

(C) $\tan 6\theta$

(D) $\cot 6\theta$

Question 10. The value of $\cos 20^\circ \cos 40^\circ \cos 80^\circ$ is:

(A) $\frac{1}{2}$

(B) $\frac{1}{4}$

(C) $\frac{1}{6}$

(D) $\frac{1}{8}$

Question 11. Simplify $\sin 10^\circ + \sin 50^\circ - \sin 70^\circ$.

(A) 0

(B) $\sin 60^\circ$

(C) $\cos 60^\circ$

(D) $\sin 10^\circ$

Question 12. If $\sin \theta = \frac{1}{2}$ and $\cos \phi = \frac{1}{2}$, where $\theta, \phi$ are acute, find $\sin(\theta + \phi)$.

(A) 0

(B) $\frac{1}{2}$

(C) $\frac{\sqrt{3}}{2}$

(D) 1

Question 13. Simplify $\sin(A+B) + \sin(A-B)$.

(A) $2 \sin A \cos B$

(B) $2 \cos A \sin B$

(C) $2 \cos A \cos B$

(D) $2 \sin A \sin B$

Question 14. Simplify $\cos(A+B) - \cos(A-B)$.

(A) $2 \sin A \cos B$

(B) $2 \cos A \sin B$

(C) $-2 \sin A \sin B$

(D) $2 \cos A \cos B$

Question 15. $\sin 5\theta - \sin 3\theta$ is equal to:

(A) $2 \cos 4\theta \sin \theta$

(B) $2 \sin 4\theta \cos \theta$

(C) $2 \cos 4\theta \cos \theta$

(D) $2 \sin 4\theta \sin \theta$

Question 16. Express $\sin x \cos y$ as sum/difference.

(A) $\frac{1}{2}[\sin(x+y) + \sin(x-y)]$

(B) $\frac{1}{2}[\sin(x+y) - \sin(x-y)]$

(C) $\frac{1}{2}[\cos(x+y) + \cos(x-y)]$

(D) $\frac{1}{2}[\cos(x-y) - \cos(x+y)]$

Question 17. $\frac{\sin A + \sin 3A}{\cos A + \cos 3A}$ is equal to:

(A) $\tan A$

(B) $\cot A$

(C) $\tan 2A$

(D) $\cot 2A$

Question 18. Simplify $\cos 10^\circ - \cos 50^\circ$.

(A) $\sin 40^\circ$

(B) $\sin 30^\circ$

(C) $\sin 20^\circ$

(D) $\sin 10^\circ$

Question 19. The value of $\frac{\cos 9^\circ + \sin 9^\circ}{\cos 9^\circ - \sin 9^\circ}$ is:

(A) $\tan 45^\circ$

(B) $\tan 54^\circ$

(C) $\tan 36^\circ$

(D) $\tan 81^\circ$

Question 20. Simplify $2 \cos 3\theta \sin \theta$.

(A) $\sin 4\theta + \sin 2\theta$

(B) $\sin 4\theta - \sin 2\theta$

(C) $\cos 4\theta + \cos 2\theta$

(D) $\cos 2\theta - \cos 4\theta$

Question 1. The principal value of the solution for $\sin x = 1$ is:

(A) 0

(B) $\frac{\pi}{2}$

(C) $\pi$

(D) $\frac{3\pi}{2}$

Question 2. The general solution of $\cos x = 0$ is:

(A) $x = n\pi, n \in \mathbb{Z}$

(B) $x = 2n\pi, n \in \mathbb{Z}$

(C) $x = (2n+1)\frac{\pi}{2}, n \in \mathbb{Z}$

(D) $x = n\frac{\pi}{2}, n \in \mathbb{Z}$

Question 3. The general solution of $\tan x = 1$ is:

(A) $x = n\pi + \frac{\pi}{4}, n \in \mathbb{Z}$

(B) $x = 2n\pi + \frac{\pi}{4}, n \in \mathbb{Z}$

(C) $x = n\pi + \frac{\pi}{2}, n \in \mathbb{Z}$

(D) $x = n\pi + \frac{\pi}{3}, n \in \mathbb{Z}$

Question 4. Find the principal solutions for $\tan x = -\sqrt{3}$.

(A) $x = \frac{2\pi}{3}, \frac{5\pi}{3}$

(B) $x = \frac{\pi}{3}, \frac{4\pi}{3}$

(C) $x = -\frac{\pi}{3}, \frac{2\pi}{3}$

(D) $x = \frac{5\pi}{6}, \frac{11\pi}{6}$

Question 5. The general solution of $\sin x = \sin \frac{\pi}{3}$ is:

(A) $x = n\pi + (-1)^n \frac{\pi}{3}, n \in \mathbb{Z}$

(B) $x = n\pi + \frac{\pi}{3}, n \in \mathbb{Z}$

(C) $x = 2n\pi + \frac{\pi}{3}, n \in \mathbb{Z}$

(D) $x = n\pi - \frac{\pi}{3}, n \in \mathbb{Z}$

Question 6. Find the general solution of $\sqrt{3} \tan x = 1$.

(A) $x = n\pi + \frac{\pi}{3}, n \in \mathbb{Z}$

(B) $x = n\pi + \frac{\pi}{6}, n \in \mathbb{Z}$

(C) $x = 2n\pi + \frac{\pi}{6}, n \in \mathbb{Z}$

(D) $x = 2n\pi + \frac{\pi}{3}, n \in \mathbb{Z}$

Question 7. The general solution of $\cos 2x = \cos \frac{\pi}{4}$ is:

(A) $x = n\pi \pm \frac{\pi}{8}, n \in \mathbb{Z}$

(B) $x = 2n\pi \pm \frac{\pi}{4}, n \in \mathbb{Z}$

(C) $x = n\pi \pm \frac{\pi}{4}, n \in \mathbb{Z}$

(D) $x = 2n\pi \pm \frac{\pi}{8}, n \in \mathbb{Z}$

Question 8. Find the number of solutions for $\sin x = \cos x$ in the interval $[0, 2\pi)$.

(A) 1

(B) 2

(C) 3

(D) 4

Question 9. The general solution of $\sin^2 x = \sin^2 \alpha$ is:

(A) $x = n\pi + (-1)^n \alpha, n \in \mathbb{Z}$

(B) $x = n\pi \pm \alpha, n \in \mathbb{Z}$

(C) $x = 2n\pi \pm \alpha, n \in \mathbb{Z}$

(D) $x = n\pi + \alpha, n \in \mathbb{Z}$

Question 10. Solve $2 \cos^2 x - 3 \cos x + 1 = 0$ for $x \in [0, 2\pi)$.

(A) $\frac{\pi}{3}, \frac{5\pi}{3}$

(B) $0, \frac{\pi}{3}, \frac{5\pi}{3}$

(C) $0, \frac{\pi}{3}, \frac{5\pi}{3}, 2\pi$

(D) $0, \pi/3, \pi, 5\pi/3, 2\pi$

Question 11. The general solution of $\tan x = \tan \frac{\pi}{6}$ is:

(A) $x = n\pi + \frac{\pi}{6}, n \in \mathbb{Z}$

(B) $x = 2n\pi + \frac{\pi}{6}, n \in \mathbb{Z}$

(C) $x = n\pi \pm \frac{\pi}{6}, n \in \mathbb{Z}$

(D) $x = n\frac{\pi}{6}, n \in \mathbb{Z}$

Question 12. Find the principal solution for $\cos x = -\frac{\sqrt{3}}{2}$.

(A) $\frac{5\pi}{6}, \frac{7\pi}{6}$

(B) $\frac{5\pi}{6}$

(C) $\frac{7\pi}{6}$

(D) $\frac{2\pi}{3}, \frac{4\pi}{3}$

Question 13. The general solution of $\tan 3x = \cot x$ is:

(A) $x = \frac{n\pi}{4} + \frac{\pi}{8}, n \in \mathbb{Z}$

(B) $x = \frac{n\pi}{4} + \frac{\pi}{4}, n \in \mathbb{Z}$

(C) $x = \frac{n\pi}{2} + \frac{\pi}{4}, n \in \mathbb{Z}$

(D) $x = \frac{n\pi}{2} + \frac{\pi}{8}, n \in \mathbb{Z}$

Question 14. Find the number of solutions for $\sin x = \frac{1}{2}$ in the interval $[0, \pi]$.

(A) 1

(B) 2

(C) 3

(D) 0

Question 15. The general solution of $\cos^2 x = \cos^2 \alpha$ is:

(A) $x = n\pi + (-1)^n \alpha, n \in \mathbb{Z}$

(B) $x = n\pi \pm \alpha, n \in \mathbb{Z}$

(C) $x = 2n\pi \pm \alpha, n \in \mathbb{Z}$

(D) $x = n\pi + \alpha, n \in \mathbb{Z}$

Question 16. Find the smallest positive value of $x$ satisfying $\sin 2x = \frac{1}{2}$.

(A) $\frac{\pi}{12}$

(B) $\frac{\pi}{6}$

(C) $\frac{\pi}{8}$

(D) $\frac{\pi}{4}$

Question 17. The equation $2 \sin x - 1 = 0$ has solutions for $x$ in the interval $[0, 2\pi)$:

(A) $\frac{\pi}{6}, \frac{5\pi}{6}$

(B) $\frac{\pi}{6}, \frac{7\pi}{6}$

(C) $\frac{\pi}{6}, \frac{11\pi}{6}$

(D) $\frac{5\pi}{6}, \frac{7\pi}{6}$

Question 18. The general solution of $\cos x = \cos \alpha$ is:

(A) $x = n\pi + (-1)^n \alpha, n \in \mathbb{Z}$

(B) $x = n\pi + \alpha, n \in \mathbb{Z}$

(C) $x = 2n\pi + \alpha, n \in \mathbb{Z}$

(D) $x = 2n\pi \pm \alpha, n \in \mathbb{Z}$

Question 19. Find the number of solutions for $\cos 2x = \frac{1}{2}$ in the interval $[0, 2\pi)$.

(A) 2

(B) 3

(C) 4

(D) 1

Question 20. The general solution of $\tan x = \tan y$ is:

(A) $x = n\pi + y, n \in \mathbb{Z}$

(B) $x = n\pi \pm y, n \in \mathbb{Z}$

(C) $x = 2n\pi + y, n \in \mathbb{Z}$

(D) $x = 2n\pi \pm y, n \in \mathbb{Z}$

Question 21. Find the principal solution for $\sin x = -1/\sqrt{2}$.

(A) $\frac{3\pi}{4}, \frac{7\pi}{4}$

(B) $\frac{5\pi}{4}, \frac{7\pi}{4}$

(C) $\frac{5\pi}{4}$

(D) $\frac{7\pi}{4}$

Question 1. The domain of $\sin^{-1} x$ is:

(A) $[- \frac{\pi}{2}, \frac{\pi}{2}]$

(B) $[-1, 1]$

(C) $(-\infty, \infty)$

(D) $[0, \pi]$

Question 2. The principal value branch of $\tan^{-1} x$ is:

(A) $[-\frac{\pi}{2}, \frac{\pi}{2}]$

(B) $(-\frac{\pi}{2}, \frac{\pi}{2})$

(C) $[0, \pi]$

(D) $(0, \pi)$

Question 3. Find the principal value of $\cos^{-1} \left(\frac{1}{2}\right)$.

(A) $\frac{\pi}{6}$

(B) $\frac{\pi}{4}$

(C) $\frac{\pi}{3}$

(D) $\frac{2\pi}{3}$

Question 4. $\tan^{-1} x + \cot^{-1} x$ is equal to:

(A) $\frac{\pi}{4}$

(B) $\frac{\pi}{2}$

(C) $\pi$

(D) $\frac{3\pi}{2}$

Question 5. What is the domain of $\tan^{-1} x$?

(A) $[-1, 1]$

(B) $(-\infty, \infty)$

(C) $[- \frac{\pi}{2}, \frac{\pi}{2}]$

(D) $[0, \pi]$

Question 6. Find the principal value of $\tan^{-1} (-1)$.

(A) $\frac{3\pi}{4}$

(B) $-\frac{\pi}{4}$

(C) $\frac{\pi}{4}$

(D) $\frac{7\pi}{4}$

Question 7. $\sin^{-1} (\sin x) = x$ is true when $x$ belongs to:

(A) $[0, \pi]$

(B) $[-\frac{\pi}{2}, \frac{\pi}{2}]$

(C) $(-\infty, \infty)$

(D) $[-1, 1]$

Question 8. $\cos^{-1} (\cos x) = x$ is true when $x$ belongs to:

(A) $[0, \pi]$

(B) $[-\frac{\pi}{2}, \frac{\pi}{2}]$

(C) $(-\infty, \infty)$

(D) $[-1, 1]$

Question 9. Find the value of $\sin^{-1} (\sin \frac{2\pi}{3})$.

(A) $\frac{2\pi}{3}$

(B) $\frac{\pi}{3}$

(C) $-\frac{2\pi}{3}$

(D) $-\frac{\pi}{3}$

Question 10. Find the value of $\cos^{-1} (\cos \frac{7\pi}{6})$.

(A) $\frac{7\pi}{6}$

(B) $\frac{\pi}{6}$

(C) $\frac{5\pi}{6}$

(D) $-\frac{\pi}{6}$

Question 11. Simplify $\sin (\cos^{-1} x)$.

(A) $\sqrt{1-x^2}$

(B) $\frac{1}{\sqrt{1-x^2}}$

(C) $\frac{x}{\sqrt{1-x^2}}$

(D) $\frac{\sqrt{1-x^2}}{x}$

Question 12. The domain of $\text{cosec}^{-1} x$ is:

(A) $[-1, 1]$

(B) $\mathbb{R} - (-1, 1)$

(C) $(-\infty, -1] \cup [1, \infty)$

(D) Both B and C

Question 13. The principal value branch of $\text{cosec}^{-1} x$ is:

(A) $[-\frac{\pi}{2}, \frac{\pi}{2}]$

(B) $[-\frac{\pi}{2}, \frac{\pi}{2}] - \{0\}$

(C) $[0, \pi]$

(D) $[0, \pi] - \{\frac{\pi}{2}\}$

Question 14. Find the value of $\sin^{-1} (-1/2)$.

(A) $\pi/6$

(B) $-\pi/6$

(C) $5\pi/6$

(D) $7\pi/6$

Question 15. $\tan^{-1} x + \tan^{-1} y = \tan^{-1} \left( \frac{x+y}{1-xy} \right)$ is valid when:

(A) $xy < 1$

(B) $xy > 1$

(C) $xy = 1$

(D) For all real $x, y$

Question 16. Find the value of $\tan (\tan^{-1} x + \cot^{-1} x)$, where $x \in \mathbb{R}$.

(A) 0

(B) 1

(C) undefined

(D) $x$

Question 17. The principal value branch of $\sec^{-1} x$ is:

(A) $[-\frac{\pi}{2}, \frac{\pi}{2}] - \{0\}$

(B) $[0, \pi] - \{\frac{\pi}{2}\}$

(C) $(-\infty, -1] \cup [1, \infty)$

(D) $[0, \pi]$

Question 18. Find the value of $\sin^{-1} (\sin 10)$, where 10 is in radians.

(A) 10

(B) $10 - 3\pi$

(C) $3\pi - 10$

(D) $10 - 2\pi$

Question 19. Simplify $\tan^{-1} \left( \frac{\cos x - \sin x}{\cos x + \sin x} \right)$, if $\frac{\pi}{4} < x < \frac{5\pi}{4}$.

(A) $\frac{\pi}{4} - x$

(B) $x - \frac{\pi}{4}$

(C) $\frac{3\pi}{4} - x$

(D) $x + \frac{\pi}{4}$

Question 20. Find the principal value of $\cot^{-1} (-\sqrt{3})$.

(A) $\frac{\pi}{6}$

(B) $-\frac{\pi}{6}$

(C) $\frac{5\pi}{6}$

(D) $\frac{7\pi}{6}$

Question 21. Which of the following is NOT a property of inverse trigonometric functions?

(A) $\sin^{-1} x + \cos^{-1} x = \pi/2$

(B) $\tan^{-1} x + \cot^{-1} x = \pi/2$

(C) $\sec^{-1} x + \text{cosec}^{-1} x = \pi/2$

(D) $\sin^{-1} (-x) = \pi - \sin^{-1} x$

Question 22. If $\tan^{-1} (2x) + \tan^{-1} (3x) = \frac{\pi}{4}$, find the value of x.

(A) $\frac{1}{6}$

(B) $\frac{1}{3}$

(C) $\frac{1}{2}$

(D) 1

Question 1. The angle of depression from an observer to an object below the horizontal level is equal to the angle of elevation from the object to the observer, assuming the observer and object are in the same vertical plane. This is due to:

(A) Alternate interior angles

(B) Corresponding angles

(C) Vertically opposite angles

(D) Co-interior angles

Question 2. A kite is flying at a height of $60\ \text{m}$ from the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is $60^\circ$. Find the length of the string, assuming that there is no slack in the string.

(A) $60\ \text{m}$

(B) $60\sqrt{3}\ \text{m}$

(C) $120\ \text{m}$

(D) $\frac{120}{\sqrt{3}}\ \text{m}$

Question 3. A $1.5\ \text{m}$ tall boy is standing at some distance from a $30\ \text{m}$ tall building. The angle of elevation from his eyes to the top of the building increases from $30^\circ$ to $60^\circ$ as he walks towards the building. Find the distance he walked towards the building.

(A) $19\sqrt{3}\ \text{m}$

(B) $28.5\sqrt{3}\ \text{m}$

(C) $19 (3 - \frac{1}{\sqrt{3}})\ \text{m}$

(D) $19 ( \frac{1}{\sqrt{3}} - \sqrt{3})\ \text{m}$

Question 4. From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a $20\ \text{m}$ high building are $45^\circ$ and $60^\circ$ respectively. Find the height of the tower.

(A) $20(\sqrt{3} - 1)\ \text{m}$

(B) $20\sqrt{3}\ \text{m}$

(C) $20\ \text{m}$

(D) $20( \sqrt{3} + 1)\ \text{m}$

Question 5. The angle of elevation of the top of a tower from a point on the ground, which is $30\ \text{m}$ away from the foot of the tower, is $30^\circ$. The height of the tower is:

(A) $10\sqrt{3}\ \text{m}$

(B) $30\sqrt{3}\ \text{m}$

(C) $\frac{30}{\sqrt{3}}\ \text{m}$

(D) $30\ \text{m}$

Question 6. A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle of $30^\circ$ with it. The distance between the foot of the tree to the point where the top touches the ground is $8\ \text{m}$. Find the height of the tree before it broke.

(A) $8\sqrt{3}\ \text{m}$

(B) $24\sqrt{3}\ \text{m}$

(C) $\frac{8\sqrt{3}}{3}\ \text{m}$

(D) $\frac{16\sqrt{3}}{3}\ \text{m}$

Question 7. An observer $1.5\ \text{m}$ tall is $28.5\ \text{m}$ away from a chimney. The angle of elevation of the top of the chimney from her eyes is $45^\circ$. What is the height of the chimney?

(A) $30\ \text{m}$

(B) $28.5\ \text{m}$

(C) $27\ \text{m}$

(D) $26\ \text{m}$

Question 8. From the top of a $7\ \text{m}$ high building, the angle of elevation of the top of a cable tower is $60^\circ$ and the angle of depression of its foot is $45^\circ$. Determine the height of the tower.

(A) $7(\sqrt{3}-1)\ \text{m}$

(B) $7(\sqrt{3}+1)\ \text{m}$

(C) $7\sqrt{3}\ \text{m}$

(D) $14\ \text{m}$

Question 9. The angle of elevation of the top of a tower from two points at a distance of $9\ \text{m}$ and $4\ \text{m}$ from the base of the tower and in the same straight line with it are complementary. Find the height of the tower.

(A) $6\ \text{m}$

(B) $6.5\ \text{m}$

(C) $7\ \text{m}$

(D) $13\ \text{m}$

Question 10. A vertical pole stands on the ground. From a point on the ground $20\ \text{m}$ away from the foot of the pole, the angle of elevation of the top of the pole is $60^\circ$. The height of the pole is:

(A) $20\sqrt{3}\ \text{m}$

(B) $\frac{20}{\sqrt{3}}\ \text{m}$

(C) $20\ \text{m}$

(D) $10\sqrt{3}\ \text{m}$

Question 11. The angles of elevation of the top of a tower from two points P and Q at distances 'a' and 'b' respectively from the base and in the same straight line with it are $\alpha$ and $\beta$. If P is closer to the tower, the height of the tower is:

(A) $\frac{b \tan \beta - a \tan \alpha}{\tan \alpha - \tan \beta}$

(B) $\frac{a \tan \alpha - b \tan \beta}{\tan \alpha - \tan \beta}$

(C) $\frac{a \tan \beta - b \tan \alpha}{\tan \alpha - \tan \beta}$

(D) $\frac{b \tan \alpha - a \tan \beta}{\tan \alpha - \tan \beta}$

Question 12. A circus artist is climbing a $20\ \text{m}$ long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. The angle made by the rope with the ground level is $30^\circ$. The height of the pole is:

(A) $10\ \text{m}$

(B) $20\ \text{m}$

(C) $10\sqrt{3}\ \text{m}$

(D) $20\sqrt{3}\ \text{m}$

Question 13. A tower is $50\ \text{m}$ high. Its shadow is $50\sqrt{3}\ \text{m}$ long when the sun's altitude is:

(A) $30^\circ$

(B) $45^\circ$

(C) $60^\circ$

(D) $90^\circ$

Question 14. From the top of a $100\ \text{m}$ high cliff, the angle of depression of a ship is $15^\circ$. If the lighthouse is $100\ \text{m}$ high, the distance of the ship from the base of the lighthouse is:

(A) $100 (\sqrt{3}+1)\ \text{m}$

(B) $100 (2+\sqrt{3})\ \text{m}$

(C) $100 (\sqrt{3}-1)\ \text{m}$

(D) $100 (2-\sqrt{3})\ \text{m}$

Question 15. A man is standing on the deck of a ship, which is $10\ \text{m}$ above water level. He observes the angle of elevation of the top of a cliff as $45^\circ$ and the angle of depression of the base of the cliff as $30^\circ$. The height of the cliff is:

(A) $10(1 + \sqrt{3})\ \text{m}$

(B) $10(1 + \frac{1}{\sqrt{3}})\ \text{m}$

(C) $10(1 + \frac{\sqrt{3}}{3})\ \text{m}$

(D) $10(1 + \frac{2}{\sqrt{3}})\ \text{m}$

Question 16. The angles of elevation of the top of a building from two points at a distance 'a' and 'b' (a>b) from the base and in the same straight line are $30^\circ$ and $60^\circ$ respectively. The height of the building is:

(A) $\sqrt{ab}$

(B) $\sqrt{\frac{a}{b}}$

(C) $\sqrt{3ab}$

(D) $\frac{\sqrt{ab}}{\sqrt{3}}$

Question 17. A flagstaff stands on the top of a $5\ \text{m}$ high building. From a point on the ground, the angle of elevation of the top of the flagstaff is $60^\circ$ and from the same point, the angle of elevation of the top of the building is $45^\circ$. Find the height of the flagstaff.

(A) $5\sqrt{3}\ \text{m}$

(B) $5(\sqrt{3}-1)\ \text{m}$

(C) $5(\sqrt{3}+1)\ \text{m}$

(D) $10\ \text{m}$

Question 18. The angle of elevation of a cloud from a point 'h' meters above a lake is $\alpha$ and the angle of depression of its reflection in the lake is $\beta$. The height of the cloud is:

(A) $\frac{h(\tan \alpha + \tan \beta)}{\tan \beta - \tan \alpha}$

(B) $\frac{h(\tan \beta - \tan \alpha)}{\tan \alpha + \tan \beta}$

(C) $\frac{h(\tan \alpha + \tan \beta)}{\tan \alpha - \tan \beta}$

(D) $\frac{h(\tan \beta + \tan \alpha)}{\tan \beta - \tan \alpha}$

Question 19. From the top of a $100\ \text{m}$ high cliff, the angles of depression of two ships are $30^\circ$ and $45^\circ$. If the ships are on the same side of the cliff and in a straight line with its foot, the distance between the two ships is:

(A) $100(\sqrt{3}-1)\ \text{m}$

(B) $100(\sqrt{3}+1)\ \text{m}$

(C) $100(2+\sqrt{3})\ \text{m}$

(D) $100\sqrt{3}\ \text{m}$

Question 20. If the angle of elevation of the top of a tower from a point is $30^\circ$, and moving $20\ \text{m}$ towards the tower, the angle of elevation becomes $60^\circ$, then the height of the tower is:

(A) $10\sqrt{3}\ \text{m}$

(B) $10\ \text{m}$

(C) $20\sqrt{3}\ \text{m}$

(D) $20\ \text{m}$