Multiple Correct Answers MCQs for Sub-Topics of Topic 8: Trigonometry

Question 1. In a right-angled triangle ABC, right-angled at B, which of the following statements are always true?

(A) AC is the longest side.

(B) $\angle A + \angle C = 90^\circ$.

(C) $AB^2 + BC^2 = AC^2$.

(D) $\tan A = \frac{BC}{AB}$.

Question 2. If $\sin \theta = \frac{p}{q}$ in a right triangle, which of the following could be true?

(A) $p < q$

(B) The opposite side is $p$ and the hypotenuse is $q$.

(C) The adjacent side is $\sqrt{q^2 - p^2}$.

(D) $\cos \theta = \frac{\sqrt{q^2 - p^2}}{q}$.

Question 3. In a right triangle, if $\tan A = 1$, which of the following are correct?

(A) The opposite side is equal to the adjacent side.

(B) $\angle A = 45^\circ$.

(C) $\sin A = \cos A$.

(D) $\sec A = \text{cosec } A$.

Question 4. Which of the following ratios are correct?

(A) $\sin \phi = \frac{o}{h}$

(B) $\cos \phi = \frac{a}{h}$

(C) $\tan \phi = \frac{o}{a}$

(D) $\text{cosec } \phi = \frac{h}{o}$.

Question 5. If $\cos \theta = \frac{12}{13}$ in a right triangle, which of the following statements are true?

(A) The adjacent side is $12$ and the hypotenuse is $13$ (in some units).

(B) The opposite side is $5$ (in the same units).

(C) $\sin \theta = \frac{5}{13}$.

(D) $\tan \theta = \frac{5}{12}$.

Question 6. Which of the following trigonometric ratios have values strictly between $0$ and $1$?

(A) $\sin \theta$

(B) $\cos \theta$

(C) $\tan \theta$

(D) $\sec \theta$

Question 7. If $\text{cosec } A = 2$, which of the following are true?

(A) $\sin A = \frac{1}{2}$.

(B) $\angle A = 30^\circ$.

(C) $\cos A = \frac{\sqrt{3}}{2}$.

(D) $\cot A = \sqrt{3}$.

Question 8. In a right triangle with angle $\alpha$, if the opposite side is $3$ units and the adjacent side is $4$ units, which of the following are correct?

(A) The hypotenuse is $5$ units.

(B) $\sin \alpha = \frac{3}{5}$.

(C) $\tan \alpha = \frac{3}{4}$.

(D) $\sec \alpha = \frac{5}{4}$.

Question 9. Which of the following ratios are reciprocals of each other?

(A) $\sin \theta$ and $\cos \theta$

(B) $\tan \theta$ and $\cot \theta$

(C) $\sec \theta$ and $\cos \theta$

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

Question 10. If in $\triangle XYZ$, right-angled at Y, XY = $15$ and XZ = $17$, which of the following are correct?

(A) YZ = $8$.

(B) $\sin X = \frac{8}{17}$.

(C) $\cos Z = \frac{8}{17}$.

(D) $\tan X = \frac{8}{15}$.

Question 11. Which of the following statements are false?

(A) $\sin \theta < 1$

(B) $\tan \theta > 1$

(C) $\sec \theta < 1$

(D) $\text{cosec } \theta > 1$

Question 12. If $\sin \theta = \cos \theta$, which of the following are true?

(A) $\tan \theta = 1$.

(B) $\theta = 45^\circ$.

(C) $\sec \theta = \sqrt{2}$.

(D) $\text{cosec } \theta = \sqrt{2}$.

Question 1. Which of the following angles have a sine value of $\frac{\sqrt{3}}{2}$?

(A) $30^\circ$

(B) $60^\circ$

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

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

Question 2. Which of the following expressions evaluate to $1$?

(A) $\sin 90^\circ$

(B) $\cos 0^\circ$

(C) $\tan 45^\circ$

(D) $\sin^2 30^\circ + \cos^2 30^\circ$

Question 3. Which of the following are equal to $\cos 30^\circ$?

(A) $\sin 60^\circ$

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

(C) $\cos (90^\circ - 60^\circ)$

(D) $\sin (90^\circ - 30^\circ)$

Question 4. If $A+B = 90^\circ$, which of the following are true?

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

(B) $\tan A = \cot B$

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

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

Question 5. Which of the following values is undefined?

(A) $\tan 90^\circ$

(B) $\cot 0^\circ$

(C) $\sec 90^\circ$

(D) $\text{cosec } 0^\circ$

Question 6. For which angle(s) is $\sin \theta = \frac{1}{\sqrt{2}}$ and $\cos \theta = \frac{1}{\sqrt{2}}$ simultaneously true?

(A) $45^\circ$

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

(C) $30^\circ$

(D) $60^\circ$

Question 7. Which of the following expressions simplify to $1$?

(A) $\frac{\sin 40^\circ}{\cos 50^\circ}$

(B) $\tan 20^\circ \cot 70^\circ$

(C) $\sin 30^\circ + \cos 60^\circ$

(D) $\sec 25^\circ - \text{cosec } 65^\circ$

Question 8. If $\tan \theta = \cot (30^\circ)$, which of the following are true?

(A) $\tan \theta = \tan 60^\circ$.

(B) $\theta = 60^\circ$.

(C) $\sin \theta = \frac{\sqrt{3}}{2}$.

(D) $\theta + 30^\circ = 90^\circ$.

Question 9. Which of the following statements about special angles are correct?

(A) $\sin 0^\circ = 0$

(B) $\cos 90^\circ = 0$

(C) $\tan 0^\circ = 0$

(D) $\cot 90^\circ = 0$

Question 10. If $\sin A = \cos B$, which of the following relationships must hold?

(A) $A+B = 90^\circ$

(B) $\tan A = \cot B$

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

(D) $A = B$ only if $A=B=45^\circ$.

Question 11. The value of $\sin 60^\circ \cos 30^\circ + \cos 60^\circ \sin 30^\circ$ is equal to which of the following?

(A) $\sin (60^\circ + 30^\circ)$

(B) $\sin 90^\circ$

(C) $1$

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

Question 12. Which of the following statements are true about the values of trigonometric ratios for special angles?

(A) $\sin 30^\circ < \sin 45^\circ$

(B) $\cos 30^\circ > \cos 45^\circ$

(C) $\tan 30^\circ < \tan 45^\circ$

(D) $\sin 60^\circ = \cos 30^\circ$

Question 1. Which of the following are Pythagorean identities?

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

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

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

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

Question 2. Which of the following expressions are equivalent to $\frac{\sin \theta}{\cos \theta}$?

(A) $\tan \theta$

(B) $\frac{1}{\cot \theta}$

(C) $\sin \theta \cdot \sec \theta$

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

Question 3. Which of the following statements are always true identities?

(A) $\sec \theta = \frac{1}{\cos \theta}$

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

(C) $\cot \theta = \frac{1}{\tan \theta}$

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

Question 4. Simplify $\sin^2 A (1 + \cot^2 A)$. Which of the following expressions are equivalent?

(A) $\sin^2 A \cdot \text{cosec}^2 A$

(B) $1$

(C) $\sin^2 A + \cos^2 A$

(D) $\tan A \cot A$

Question 5. If $\sec \theta + \tan \theta = k$, which of the following are correct?

(A) $\sec \theta - \tan \theta = \frac{1}{k}$.

(B) $\sec \theta = \frac{1}{2}(k + \frac{1}{k})$.

(C) $\tan \theta = \frac{1}{2}(k - \frac{1}{k})$.

(D) $\sin \theta = \frac{k^2-1}{k^2+1}$.

Question 6. Simplify $\sin \theta \cdot \cot \theta$. Which of the following expressions simplify to $\cos \theta$?

(A) $\sin \theta \cdot \cot \theta$

(B) $\frac{\sin \theta}{\tan \theta}$

(C) $\sqrt{1 - \sin^2 \theta}$

(D) $\frac{1}{\sec \theta}$

Question 7. Simplify $\frac{1 + \tan^2 A}{1 + \cot^2 A}$. Which of the following are equivalent?

(A) $\tan^2 A$

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

(C) $\left(\frac{\sin A}{\cos A}\right)^2$

(D) $\frac{1-\cos^2 A}{1-\sin^2 A}$

Question 8. Which of the following are identities?

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

(B) $(\sec \theta - \tan \theta)^2 = \frac{1 - \sin \theta}{1 + \sin \theta}$

(C) $\sin^4 \theta - \cos^4 \theta = \sin^2 \theta - \cos^2 \theta$

(D) $\tan^2 \theta - \sin^2 \theta = \tan^2 \theta \sin^2 \theta$

Question 9. If $\sin \theta + \text{cosec } \theta = 2$, which of the following must be true?

(A) $\sin \theta = 1$.

(B) $\text{cosec } \theta = 1$.

(C) $\sin^2 \theta + \text{cosec}^2 \theta = 2$.

(D) $\sin^n \theta + \text{cosec}^n \theta = 2$ for any integer n.

Question 10. The expression $\frac{\cos A}{1 - \sin A}$ is equivalent to which of the following?

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

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

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

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

Question 11. Which of the following identities involve reciprocal ratios?

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

(B) $\cos \theta = \frac{1}{\sec \theta}$

(C) $\tan \theta = \frac{1}{\cot \theta}$

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

Question 12. If $\cos \theta - \sin \theta = \sqrt{2} \sin \theta$, which of the following are correct?

(A) $\cos \theta = (\sqrt{2}+1) \sin \theta$.

(B) $\cot \theta = \sqrt{2}+1$.

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

(D) $\frac{\cos \theta + \sin \theta}{\cos \theta - \sin \theta} = \sqrt{2}$.

Question 1. Which of the following angle measures are equivalent to $90^\circ$?

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

(B) $\frac{1}{4}$ of a complete angle

(C) $5400'$ (minutes)

(D) $324000''$ (seconds)

Question 2. The value of $\pi$ radians is equivalent to which of the following?

(A) $180^\circ$

(B) A straight angle

(C) $\frac{1}{2}$ of a complete angle in radians

(D) Approximately $3.14159$ radians

Question 3. To convert an angle from degrees to radians, you can multiply the degree measure by which of the following factors?

(A) $\frac{\pi}{180}$

(B) $\frac{180}{\pi}$

(C) $\frac{\text{circumference}}{\text{diameter}}$ divided by $180$

(D) The ratio of $\pi$ to the number of degrees in a straight angle.

Question 4. Which of the following pairs of angle measures are equivalent?

(A) $30^\circ$ and $\frac{\pi}{6}$ radians

(B) $45^\circ$ and $\frac{\pi}{4}$ radians

(C) $60^\circ$ and $\frac{\pi}{3}$ radians

(D) $120^\circ$ and $\frac{2\pi}{3}$ radians

Question 5. If an arc of length 'l' subtends a central angle $\theta$ (in radians) in a circle of radius 'r', which of the following formulas are correct?

(A) $l = r\theta$

(B) $\theta = \frac{l}{r}$

(C) $r = \frac{l}{\theta}$

(D) Area of sector = $\frac{1}{2} r^2 \theta$

Question 6. Convert $225^\circ$ into radians. Which of the following are correct representations?

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

(B) $\frac{225}{180} \pi$ radians

(C) $1.25 \pi$ radians

(D) $\frac{9\pi}{4}$ radians

Question 7. Convert $\frac{4\pi}{3}$ radians into degrees. Which of the following are correct?

(A) $240^\circ$

(B) $\frac{4 \times 180}{3}$ degrees

(C) $4 \times 60^\circ$

(D) $4 \times 30^\circ$

Question 8. A wheel rotates at 120 revolutions per minute. Which of the following statements about its angular speed are correct?

(A) It makes $2$ revolutions per second.

(B) It turns through $4\pi$ radians in one second.

(C) Its angular speed is $4\pi$ radians per second.

(D) It turns through $120 \times 2\pi$ radians in one minute.

Question 9. If an arc of length $11\ \text{cm}$ subtends an angle of $30^\circ$ at the centre of a circle, find the radius of the circle. Which of the following are correct intermediate steps or the final answer?

(A) Convert $30^\circ$ to radians: $\frac{\pi}{6}$ radians.

(B) Use the formula $l = r\theta$.

(C) $11 = r \cdot \frac{22}{7} \cdot \frac{1}{6}$.

(D) $r = 21\ \text{cm}$.

Question 10. Which of the following angles are co-terminal with $30^\circ$?

(A) $390^\circ$

(B) $-330^\circ$

(C) $750^\circ$

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

Question 11. What is the area of a sector of a circle with radius $14\ \text{cm}$ and central angle $45^\circ$? Which of the following are correct?

(A) The central angle in radians is $\frac{\pi}{4}$.

(B) The area of the sector is given by $\frac{1}{2} r^2 \theta_{radians}$.

(C) Area = $\frac{1}{2} \times 14^2 \times \frac{22}{7} \times \frac{1}{4}\ \text{cm}^2$.

(D) Area = $77\ \text{cm}^2$.

Question 12. One radian is approximately equal to:

(A) $57.3^\circ$

(B) $57^\circ 17' 45''$

(C) The angle subtended by an arc equal to the radius.

(D) $180/\pi$ degrees.

Question 1. If the terminal side of an angle $\theta$ intersects the unit circle at the point $(x, y)$, which of the following are correct?

(A) $\cos \theta = x$

(B) $\sin \theta = y$

(C) $\tan \theta = y/x$

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

Question 2. In which of the following quadrants is $\sin \theta$ positive?

(A) Quadrant I

(B) Quadrant II

(C) Quadrant III

(D) Quadrant IV

Question 3. In which of the following quadrants is $\tan \theta$ positive?

(A) Quadrant I

(B) Quadrant II

(C) Quadrant III

(D) Quadrant IV

Question 4. Which of the following trigonometric functions have a range of $[-1, 1]$?

(A) $\sin x$

(B) $\cos x$

(C) $\tan x$

(D) $\sec x$

Question 5. Which of the following angles results in a point on the unit circle with a negative x-coordinate?

(A) $120^\circ$

(B) $210^\circ$

(C) $300^\circ$

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

Question 6. Which of the following trigonometric functions have a period of $2\pi$?

(A) $\sin x$

(B) $\cos x$

(C) $\sec x$

(D) $\text{cosec } x$

Question 7. If $\theta$ is in the third quadrant, which of the following trigonometric functions are positive?

(A) $\sin \theta$

(B) $\cos \theta$

(C) $\tan \theta$

(D) $\cot \theta$

Question 8. The domain of $\tan x$ includes all real numbers except angles where the terminal side lies on which axis?

(A) Positive x-axis

(B) Negative x-axis

(C) Positive y-axis

(D) Negative y-axis

Question 9. The range of $\sec x$ is:

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

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

(C) All real numbers except those strictly between $-1$ and $1$.

(D) $[-1, 1]$

Question 10. Which of the following statements about the periodicity of trigonometric functions are correct?

(A) The period of $\sin x$ is $2\pi$.

(B) The period of $\cos x$ is $2\pi$.

(C) The period of $\tan x$ is $\pi$.

(D) The period of $\cot x$ is $\pi$.

Question 11. If $\sin \theta = 0$, which of the following can be the value of $\theta$?

(A) $0$ radians

(B) $\pi$ radians

(C) $180^\circ$

(D) $360^\circ$

Question 12. If $\cos \theta = -1$, which of the following can be the value of $\theta$?

(A) $\pi$ radians

(B) $180^\circ$

(C) $3\pi$ radians

(D) $-180^\circ$

Question 1. The graph of $y = \sin x$ has:

(A) A period of $2\pi$.

(B) A maximum value of $1$.

(C) Roots (x-intercepts) at $x = n\pi$, where $n$ is an integer.

(D) Symmetry about the origin.

Question 2. The graph of $y = \cos x$ has:

(A) A period of $2\pi$.

(B) A minimum value of $-1$.

(C) y-intercept at $(0, 1)$.

(D) Symmetry about the y-axis.

Question 3. The graph of $y = \tan x$ has:

(A) A period of $\pi$.

(B) Vertical asymptotes at $x = (2n+1)\frac{\pi}{2}$, where $n$ is an integer.

(C) Roots (x-intercepts) at $x = n\pi$, where $n$ is an integer.

(D) A range of $(-\infty, \infty)$.

Question 4. Which of the following graphs have a period of $\pi$?

(A) $y = \sin (2x)$

(B) $y = \cos (2x)$

(C) $y = \tan x$

(D) $y = \cot x$

Question 5. The graph of $y = \sec x$ has vertical asymptotes at values of $x$ where:

(A) $\cos x = 0$

(B) The terminal side of $x$ lies on the positive or negative y-axis.

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

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

Question 6. The graph of $y = \text{cosec } x$ has a range of:

(A) $(-\infty, \infty)$

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

(C) $\{y \in \mathbb{R} : |y| \geq 1\}$

(D) $[-1, 1]$

Question 7. What is the amplitude of the function $y = 5 \cos (3x)$?

(A) $5$

(B) $-5$

(C) $|5|$

(D) The distance from the midline to the maximum or minimum value.

Question 8. The graph of $y = \cot x$ has:

(A) A period of $\pi$.

(B) Vertical asymptotes at $x = n\pi$, where $n$ is an integer.

(C) Roots (x-intercepts) at $x = (2n+1)\frac{\pi}{2}$, where $n$ is an integer.

(D) Symmetry about the origin.

Question 9. Which of the following statements are true about the graph of $y = \sin x$ compared to $y = \cos x$?

(A) Both have the same period.

(B) Both have the same amplitude.

(C) The graph of $\cos x$ is a phase shift of the graph of $\sin x$.

(D) The graph of $\sin x$ passes through $(0,0)$ while $\cos x$ passes through $(0,1)$.

Question 10. The range of $y = 2 - \sin x$ is:

(A) $[1, 3]$

(B) $1 \leq y \leq 3$

(C) $[2-1, 2+1]$

(D) $[-1, 1]$

Question 11. Which of the following functions are periodic?

(A) $y = \sin x$

(B) $y = x^2$

(C) $y = \tan x$

(D) $y = \sec x$

Question 12. The vertical asymptotes of the graph of $y = \tan x$ occur at angles $\theta$ where:

(A) $\cos \theta = 0$

(B) The tangent function is undefined.

(C) The terminal side of $\theta$ is along the positive or negative y-axis.

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

Question 1. Which of the following are correct formulas for $\sin(A+B)$?

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

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

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

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

Question 2. Which of the following are correct formulas for $\cos(A-B)$?

(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 3. Which of the following expressions are equal to $\cos 2A$?

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

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

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

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

Question 4. The value of $\sin 15^\circ$ is equal to the value of which of the following?

(A) $\sin (45^\circ - 30^\circ)$

(B) $\sin (60^\circ - 45^\circ)$

(C) $\cos 75^\circ$

(D) $\frac{\sqrt{6} - \sqrt{2}}{4}$

Question 5. Which of the following are correct formulas for $\tan 2A$?

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

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

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

(D) $\frac{2 \sin A \cos A}{\cos^2 A - \sin^2 A}$

Question 6. If $\tan A = 2$ and $\tan B = 3$, which of the following are true about $A+B$?

(A) $\tan(A+B) = -1$.

(B) $A+B = \frac{3\pi}{4}$.

(C) $A+B = 135^\circ$.

(D) The sum of two acute angles with tangents $2$ and $3$ is an obtuse angle.

Question 7. Which of the following expressions are equivalent to $\sin 2A$?

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

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

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

(D) $2 \sin A$

Question 8. If $\cos A = x$, which of the following can be expressions for $\cos 2A$ in terms of $x$?

(A) $2x^2 - 1$

(B) $1 - 2(1-x^2)$

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

(D) $x^2 - (1-x^2)$

Question 9. Which of the following identities are related to half-angle formulas?

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

(B) $\cos^2 \frac{\theta}{2} = \frac{1 + \cos \theta}{2}$

(C) $\tan^2 \frac{\theta}{2} = \frac{1 - \cos \theta}{1 + \cos \theta}$

(D) $\tan \frac{\theta}{2} = \frac{1 - \cos \theta}{\sin \theta}$

Question 10. If $\cos \theta = \frac{4}{5}$, and $\theta$ is acute, which of the following are correct values?

(A) $\sin \theta = \frac{3}{5}$

(B) $\sin 2\theta = \frac{24}{25}$

(C) $\cos 2\theta = \frac{7}{25}$

(D) $\tan 2\theta = \frac{24}{7}$

Question 11. Which of the following expressions are equivalent to $\frac{1 - \cos 2\theta}{\sin 2\theta}$?

(A) $\tan \theta$

(B) $\frac{2 \sin^2 \theta}{2 \sin \theta \cos \theta}$

(C) $\cot \theta$

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

Question 12. Which of the following are applications of sum and difference identities?

(A) Finding the value of $\sin 75^\circ$.

(B) Finding the value of $\tan 15^\circ$.

(C) Proving identities like $\sin(A+B)\sin(A-B) = \sin^2 A - \sin^2 B$.

(D) Deriving double angle formulas.

Question 1. Which of the following are correct product-to-sum formulas?

(A) $2 \sin A \cos B = \sin(A+B) + \sin(A-B)$

(B) $2 \cos A \sin B = \sin(A+B) - \sin(A-B)$

(C) $2 \cos A \cos B = \cos(A+B) + \cos(A-B)$

(D) $2 \sin A \sin B = \cos(A-B) - \cos(A+B)$

Question 2. Which of the following are correct sum-to-product formulas?

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

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

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

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

Question 3. Express $2 \sin 5\theta \cos 3\theta$ as a sum or difference. Which of the following are equivalent?

(A) $\sin(5\theta + 3\theta) + \sin(5\theta - 3\theta)$

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

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

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

Question 4. Express $\sin 7x - \sin 3x$ as a product. Which of the following are equivalent?

(A) $2 \cos \left(\frac{7x+3x}{2}\right) \sin \left(\frac{7x-3x}{2}\right)$

(B) $2 \cos 5x \sin 2x$

(C) $2 \sin 5x \cos 2x$

(D) $\cos 10x - \cos 4x$

Question 5. Which of the following identities can be derived using sum-to-product or product-to-sum formulas?

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

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

(C) $\frac{\sin x + \sin y}{\cos x + \cos y} = \tan \left(\frac{x+y}{2}\right)$

(D) $\cos^2 \theta = \frac{1 + \cos 2\theta}{2}$

Question 6. Simplify $\cos 5\theta + \cos 3\theta$. Which of the following are equivalent?

(A) $2 \cos \left(\frac{5\theta+3\theta}{2}\right) \cos \left(\frac{5\theta-3\theta}{2}\right)$

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

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

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

Question 7. The expression $\sin 20^\circ \sin 40^\circ \sin 80^\circ$ is equal to which of the following?

(A) $\frac{1}{4} \sin (3 \times 20^\circ)$

(B) $\frac{1}{4} \sin 60^\circ$

(C) $\frac{1}{4} \frac{\sqrt{3}}{2}$

(D) $\frac{\sqrt{3}}{8}$

Question 8. Simplify $\frac{\sin A + \sin B}{\cos A - \cos B}$. Which of the following are equivalent (where defined)?

(A) $\cot \left(\frac{A-B}{2}\right)$

(B) $-\cot \left(\frac{B-A}{2}\right)$

(C) $-\cot \left(\frac{A-B}{2}\right)$

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

Question 9. Which of the following are useful applications of trigonometric transformations?

(A) Converting products of trigonometric functions into sums or differences.

(B) Converting sums or differences of trigonometric functions into products.

(C) Simplifying trigonometric expressions.

(D) Solving certain types of trigonometric equations.

Question 10. Simplify $2 \cos A \cos B$. Which of the following are equivalent?

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

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

(C) The sum of $\cos(A+B)$ and $\cos(A-B)$.

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

Question 11. The expression $\sin 10^\circ + \sin 20^\circ + \sin 30^\circ + ... + \sin 180^\circ$ involves sums of sines. Which methods could be useful in evaluating such a sum?

(A) Pairing terms using sum-to-product formulas.

(B) Looking for terms that cancel out.

(C) Using the property $\sin(\pi - x) = \sin x$.

(D) Using the property $\sin(2\pi - x) = -\sin x$.

Question 12. Which of the following are correct transformations?

(A) $\sin x + \sin y = 2 \sin(\frac{x+y}{2}) \cos(\frac{x-y}{2})$

(B) $\cos x - \cos y = -2 \sin(\frac{x+y}{2}) \sin(\frac{x-y}{2})$

(C) $2 \sin A \sin B = \cos(A-B) - \cos(A+B)$

(D) $2 \cos A \sin B = \sin(A+B) - \sin(A-B)$

Question 1. Which of the following are solutions to the equation $\sin x = \frac{\sqrt{3}}{2}$ in the interval $[0, 2\pi)$?

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

(B) $60^\circ$

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

(D) $120^\circ$

Question 2. The general solution for $\cos x = \cos \alpha$ is $x = 2n\pi \pm \alpha$, where $n \in \mathbb{Z}$. This implies that if $\cos x = \frac{1}{2}$, the general solution is:

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

(B) $x = 2n\pi + \frac{\pi}{3}$ for some integer $n$

(C) $x = 2n\pi - \frac{\pi}{3}$ for some integer $n$

(D) $x = 2n\pi \pm 60^\circ, n \in \mathbb{Z}$

Question 3. Which of the following are part of the general solution for $\tan x = 1$?

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

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

(C) $\frac{9\pi}{4}$

(D) $45^\circ + n \times 180^\circ, n \in \mathbb{Z}$

Question 4. Which of the following equations have $x=0$ as a solution?

(A) $\sin x = 0$

(B) $\cos x = 1$

(C) $\tan x = 0$

(D) $\sin x + \cos x = 1$

Question 5. The equation $2 \sin^2 x + \sin x - 1 = 0$ can be factored. Which of the following are correct factorizations or intermediate steps?

(A) $(2 \sin x - 1)(\sin x + 1) = 0$

(B) $2y^2 + y - 1 = 0$, where $y = \sin x$

(C) $\sin x = 1/2$ or $\sin x = -1$

(D) The solutions for $\sin x = 1/2$ are $\frac{\pi}{6}, \frac{5\pi}{6}$.

Question 6. Which of the following equations have a general solution of the form $x = n\pi, n \in \mathbb{Z}$?

(A) $\sin x = 0$

(B) $\tan x = 0$

(C) $\cot x$ is undefined

(D) $\cos x = 1$

Question 7. The general solution for $\tan x = \tan \alpha$ is $x = n\pi + \alpha$, where $n \in \mathbb{Z}$. Which of the following are correct applications of this formula?

(A) If $\tan x = \sqrt{3}$, then $x = n\pi + \frac{\pi}{3}$.

(B) If $\tan x = -1$, then $x = n\pi - \frac{\pi}{4}$.

(C) If $\tan x = -1$, then $x = n\pi + \frac{3\pi}{4}$.

(D) If $\tan x = 0$, then $x = n\pi$.

Question 8. Consider the equation $\sin 2x = \sin x$. Which of the following are correct steps or solutions?

(A) $\sin 2x - \sin x = 0$

(B) $2 \cos(\frac{3x}{2}) \sin(\frac{x}{2}) = 0$

(C) $\cos(\frac{3x}{2}) = 0$ or $\sin(\frac{x}{2}) = 0$

(D) Solutions are $x = n\pi$ or $x = (2k+1)\frac{\pi}{3}$, where $n, k \in \mathbb{Z}$.

Question 9. The general solution of $\sin x = k$, where $-1 \leq k \leq 1$, is $x = n\pi + (-1)^n \alpha$, where $\alpha = \sin^{-1} k$ is the principal value. Which of the following are correct applications?

(A) If $\sin x = 1$, then $\alpha = \pi/2$, and $x = n\pi + (-1)^n \frac{\pi}{2}$.

(B) If $\sin x = -1$, then $\alpha = -\pi/2$, and $x = n\pi + (-1)^n (-\frac{\pi}{2})$.

(C) If $\sin x = 0$, then $\alpha = 0$, and $x = n\pi$.

(D) If $\sin x = 1/\sqrt{2}$, then $\alpha = \pi/4$, and $x = n\pi + (-1)^n \frac{\pi}{4}$.

Question 10. Consider the equation $\tan^2 x = 3$. Which of the following are correct?

(A) $\tan x = \sqrt{3}$ or $\tan x = -\sqrt{3}$.

(B) For $\tan x = \sqrt{3}$, the general solution is $x = n\pi + \frac{\pi}{3}$.

(C) For $\tan x = -\sqrt{3}$, the general solution is $x = n\pi - \frac{\pi}{3}$.

(D) The combined general solution can be written as $x = n\pi \pm \frac{\pi}{3}$, where $n \in \mathbb{Z}$.

Question 11. Which of the following statements about principal solutions of trigonometric equations are true?

(A) For $\sin x = k$, the principal solution $\alpha$ lies in $[-\pi/2, \pi/2]$.

(B) For $\cos x = k$, the principal solution $\alpha$ lies in $[0, \pi]$.

(C) For $\tan x = k$, the principal solution $\alpha$ lies in $(-\pi/2, \pi/2)$.

(D) Principal solutions are unique for a given value of k.

Question 12. Solve the equation $\cos 2x = \sin x$. Which of the following are correct steps or solutions?

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

(B) $2\sin^2 x + \sin x - 1 = 0$

(C) $(\sin x + 1)(2 \sin x - 1) = 0$

(D) $\sin x = -1$ or $\sin x = 1/2$.

Question 1. For the function $y = \sin^{-1} x$, which of the following statements are true?

(A) Its domain is $[-1, 1]$.

(B) Its range is $[-\frac{\pi}{2}, \frac{\pi}{2}]$.

(C) It is the inverse of the function $f(x) = \sin x$ restricted to the domain $[-\frac{\pi}{2}, \frac{\pi}{2}]$.

(D) $\sin^{-1} (\sin x) = x$ for all $x \in \mathbb{R}$.

Question 2. For the function $y = \cos^{-1} x$, which of the following statements are true?

(A) Its domain is $[-1, 1]$.

(B) Its range is $[0, \pi]$.

(C) It is the inverse of the function $f(x) = \cos x$ restricted to the domain $[0, \pi]$.

(D) $\cos^{-1} (\cos x) = x$ for all $x \in [0, \pi]$.

Question 3. Which of the following are correct principal values?

(A) $\sin^{-1} (\frac{1}{2}) = \frac{\pi}{6}$

(B) $\cos^{-1} (\frac{\sqrt{3}}{2}) = \frac{\pi}{6}$

(C) $\tan^{-1} (1) = \frac{\pi}{4}$

(D) $\sin^{-1} (-\frac{1}{\sqrt{2}}) = -\frac{\pi}{4}$

Question 4. Which of the following identities are correct?

(A) $\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}$, for $x \in [-1, 1]$.

(B) $\tan^{-1} x + \cot^{-1} x = \frac{\pi}{2}$, for $x \in \mathbb{R}$.

(C) $\sec^{-1} x + \text{cosec}^{-1} x = \frac{\pi}{2}$, for $x \in \mathbb{R} - (-1, 1)$.

(D) $\sin^{-1} (-x) = -\sin^{-1} x$, for $x \in [-1, 1]$.

Question 5. Which of the following are correct domain specifications for the inverse trigonometric functions?

(A) Domain of $\sin^{-1} x$ is $[-1, 1]$.

(B) Domain of $\cos^{-1} x$ is $[-1, 1]$.

(C) Domain of $\tan^{-1} x$ is $\mathbb{R}$.

(D) Domain of $\sec^{-1} x$ is $(-\infty, -1] \cup [1, \infty)$.

Question 6. Simplify $\sin (\tan^{-1} x)$. Which of the following expressions are equivalent?

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

(B) If $\theta = \tan^{-1} x$, then $\tan \theta = x = x/1$.

(C) Consider a right triangle with opposite side $x$ and adjacent side $1$.

(D) The hypotenuse of such a triangle is $\sqrt{1+x^2}$.

Question 7. Find the value of $\cos^{-1} (\cos \frac{5\pi}{4})$. Which of the following are correct steps or the final answer?

(A) The principal value branch of $\cos^{-1} x$ is $[0, \pi]$.

(B) $\frac{5\pi}{4}$ is not in the principal value branch.

(C) $\cos \frac{5\pi}{4} = -\frac{1}{\sqrt{2}}$.

(D) $\cos^{-1} (-\frac{1}{\sqrt{2}}) = \frac{3\pi}{4}$.

Question 8. Which of the following properties for inverse trigonometric functions hold?

(A) $\cos^{-1} (-x) = \pi - \cos^{-1} x$, for $x \in [-1, 1]$.

(B) $\tan^{-1} (-x) = -\tan^{-1} x$, for $x \in \mathbb{R}$.

(C) $\cot^{-1} (-x) = \pi - \cot^{-1} x$, for $x \in \mathbb{R}$.

(D) $\sec^{-1} (-x) = \pi + \sec^{-1} x$, for $x \in \mathbb{R} - (-1, 1)$.

Question 9. Simplify $\tan^{-1} (x) + \tan^{-1} (y)$. Which of the following formulas are correct under appropriate conditions?

(A) $\tan^{-1} \left( \frac{x+y}{1-xy} \right)$, if $xy < 1$.

(B) $\pi + \tan^{-1} \left( \frac{x+y}{1-xy} \right)$, if $xy > 1$ and $x, y > 0$.

(C) $-\pi + \tan^{-1} \left( \frac{x+y}{1-xy} \right)$, if $xy > 1$ and $x, y < 0$.

(D) $\tan^{-1} \left( \frac{x-y}{1+xy} \right)$

Question 10. Find the value of $\tan (\sin^{-1} (\frac{3}{5}) + \cos^{-1} (\frac{3}{5}))$. Which of the following are correct steps or the final answer?

(A) Let $A = \sin^{-1} (\frac{3}{5})$ and $B = \cos^{-1} (\frac{3}{5})$.

(B) $A+B = \frac{\pi}{2}$.

(C) We need to find $\tan (\frac{\pi}{2})$.

(D) The value is undefined.

Question 11. Which of the following statements about the range of inverse trigonometric functions are correct?

(A) Range of $\sin^{-1} x$ is $[-\frac{\pi}{2}, \frac{\pi}{2}]$.

(B) Range of $\tan^{-1} x$ is $(-\frac{\pi}{2}, \frac{\pi}{2})$.

(C) Range of $\cot^{-1} x$ is $(0, \pi)$.

(D) Range of $\text{cosec}^{-1} x$ is $[-\frac{\pi}{2}, \frac{\pi}{2}] - \{0\}$.

Question 12. Find the value of $\sin (\cos^{-1} (\frac{4}{5}))$. Which of the following are correct steps or the final answer?

(A) Let $\theta = \cos^{-1} (\frac{4}{5})$. Then $\cos \theta = \frac{4}{5}$.

(B) We need to find $\sin \theta$.

(C) Using $\sin^2 \theta + \cos^2 \theta = 1$, $\sin^2 \theta = 1 - (\frac{4}{5})^2 = 1 - \frac{16}{25} = \frac{9}{25}$.

(D) Since $\cos^{-1} (\frac{4}{5})$ is in $[0, \pi]$, $\theta$ is acute, so $\sin \theta = \sqrt{\frac{9}{25}} = \frac{3}{5}$.

Question 1. The angle of elevation is defined as the angle between the horizontal line and the line of sight when the observer is looking upwards. Which of the following are correct statements regarding this?

(A) It is measured upwards from the horizontal line.

(B) It is used when calculating the height of an object from a distance.

(C) The angle of elevation from point A to point B is equal to the angle of depression from point B to point A.

(D) It is always an acute angle in typical heights and distances problems.

Question 2. A tower stands vertically on the ground. From a point on the ground, which is $15\ \text{m}$ away from the foot of the tower, the angle of elevation of the top of the tower is $60^\circ$. Which of the following are correct steps or the height of the tower?

(A) Let the height of the tower be h.

(B) $\tan 60^\circ = \frac{h}{15}$.

(C) $\sqrt{3} = \frac{h}{15}$.

(D) $h = 15\sqrt{3}\ \text{m}$.

Question 3. From the top of a $60\ \text{m}$ high building, the angle of depression of an object on the ground is $30^\circ$. Which of the following are correct steps or the distance of the object from the foot of the building?

(A) The angle of elevation from the object to the top of the building is $30^\circ$.

(B) Let the distance of the object from the foot of the building be x.

(C) $\tan 30^\circ = \frac{60}{x}$.

(D) $x = 60\sqrt{3}\ \text{m}$.

Question 4. A kite is flying at a height of $75\ \text{m}$ from the ground. The string attached to the kite is inclined at $60^\circ$ to the ground. Which of the following are correct steps or the length of the string?

(A) Let the length of the string be L.

(B) $\sin 60^\circ = \frac{75}{L}$.

(C) $\frac{\sqrt{3}}{2} = \frac{75}{L}$.

(D) $L = \frac{150}{\sqrt{3}} = 50\sqrt{3}\ \text{m}$.

Question 5. From a point P on the ground, the angle of elevation of the top of a tower is $30^\circ$. From a point Q on the line segment joining P to the foot of the tower, the angle of elevation of the top of the tower is $60^\circ$. If PQ = $40\ \text{m}$, let h be the height of the tower and x be the distance of Q from the foot of the tower. Which of the following equations or statements are correct?

(A) $\tan 30^\circ = \frac{h}{40+x}$.

(B) $\tan 60^\circ = \frac{h}{x}$.

(C) $\frac{1}{\sqrt{3}} = \frac{h}{40+x}$ and $\sqrt{3} = \frac{h}{x}$.

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

Question 6. A man standing on a ship observes a cliff. The angle of elevation of the top of the cliff is $45^\circ$. If the man moves $100\ \text{m}$ towards the cliff, the angle of elevation becomes $60^\circ$. Let h be the height of the cliff. Which of the following equations or statements are correct?

(A) Let the initial distance from the foot of the cliff be x.

(B) $\tan 45^\circ = \frac{h}{x}$ and $\tan 60^\circ = \frac{h}{x-100}$.

(C) $h = x$ and $\sqrt{3} = \frac{h}{h-100}$.

(D) $h = 50(\sqrt{3}+1)\ \text{m}$.

Question 7. The angles of elevation of the top of a tower from two points at a distance 'a' and 'b' from the base and in the same straight line with it are complementary ($\alpha$ and $90^\circ-\alpha$). If the points are on the same side of the tower, and 'a' is closer, which of the following are correct?

(A) $\tan \alpha = h/a$ and $\tan(90^\circ-\alpha) = h/b$.

(B) $\tan \alpha = h/a$ and $\cot \alpha = h/b$.

(C) $(h/a) \cdot (h/b) = 1$.

(D) $h^2 = ab$, so $h = \sqrt{ab}$.

Question 8. From the top of a building, the angle of depression of a car parked on the road is $45^\circ$. If the distance of the car from the building is $25\ \text{m}$, which of the following are correct?

(A) The angle of elevation from the car to the top of the building is $45^\circ$.

(B) The height of the building is equal to the distance of the car from the building.

(C) The height of the building is $25\ \text{m}$.

(D) $\tan 45^\circ = \frac{\text{Height}}{\text{Distance}}$.

Question 9. A bridge across a river makes an angle of $45^\circ$ with the bank. If the length of the bridge across the river is $150\ \text{m}$, which of the following are correct statements about the width of the river?

(A) Let the width of the river be w.

(B) $\sin 45^\circ = \frac{w}{150}$.

(C) $\frac{1}{\sqrt{2}} = \frac{w}{150}$.

(D) $w = \frac{150}{\sqrt{2}} = 75\sqrt{2}\ \text{m}$.

Question 10. The shadow of a tower standing on a level ground is found to be $40\ \text{m}$ longer when the sun's altitude is $30^\circ$ than when it is $60^\circ$. Let h be the height of the tower. Which of the following equations can be used to find h?

(A) $\tan 60^\circ = \frac{h}{x}$.

(B) $\tan 30^\circ = \frac{h}{x+40}$.

(C) $\sqrt{3} = h/x$ and $1/\sqrt{3} = h/(x+40)$.

(D) $h(\sqrt{3} - 1/\sqrt{3}) = 40$.

Question 11. An aeroplane flying at a height of $300\ \text{m}$ above the ground passes vertically above another aeroplane at an instant when the angles of elevation of the two aeroplanes from the same point on the ground are $60^\circ$ and $45^\circ$ respectively. Let the height of the lower aeroplane be h. Which of the following are correct?

(A) Let the distance of the observation point from the point directly below the aeroplanes be x.

(B) $\tan 60^\circ = \frac{300}{x}$.

(C) $\tan 45^\circ = \frac{h}{x}$.

(D) $x = 300/\sqrt{3} = 100\sqrt{3}\ \text{m}$.

Question 12. A statue $1.6\ \text{m}$ tall stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is $60^\circ$ and from the same point the angle of elevation of the top of the pedestal is $45^\circ$. Let h be the height of the pedestal and x be the distance of the point from the pedestal. Which of the following are correct?

(A) $\tan 45^\circ = h/x$, so $h=x$.

(B) $\tan 60^\circ = \frac{h+1.6}{x}$.

(C) $\sqrt{3} = \frac{h+1.6}{h}$.

(D) $h = \frac{1.6}{\sqrt{3}-1} = 0.8(\sqrt{3}+1)\ \text{m}$.