Completing Statements MCQs for Sub-Topics of Topic 8: Trigonometry

Question 1. In a right-angled triangle, the side opposite the right angle is always the____

(A) Opposite side

(B) Adjacent side

(C) Hypotenuse

(D) Base

Question 2. In a right-angled triangle, the sum of the two acute angles is____

(A) $45^\circ$

(B) $90^\circ$

(C) $180^\circ$

(D) $360^\circ$

Question 3. The ratio of the length of the side opposite an acute angle to the length of the hypotenuse in a right triangle defines the trigonometric ratio called____

(A) Cosine

(B) Sine

(C) Tangent

(D) Secant

Question 4. For an acute angle $\theta$ in a right triangle, $\cos \theta$ is the ratio of the adjacent side to the____

(A) Opposite side

(B) Adjacent side

(C) Hypotenuse

(D) Perpendicular

Question 5. If the length of the side opposite an acute angle in a right triangle is 'o' and the length of the adjacent side is 'a', then $\tan \theta$ is given by____

(A) $o/a$

(B) $a/o$

(C) $o/h$ (where h is hypotenuse)

(D) $a/h$ (where h is hypotenuse)

Question 6. The reciprocal of $\sin \theta$ is defined as____

(A) $\cos \theta$

(B) $\sec \theta$

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

(D) $\cot \theta$

Question 7. $\frac{1}{\cos \theta}$ is defined as the trigonometric ratio called____

(A) Sine

(B) Cosine

(C) Tangent

(D) Secant

Question 8. The ratio of the length of the adjacent side to the length of the opposite side for an acute angle in a right triangle is the____

(A) Tangent

(B) Cotangent

(C) Secant

(D) Cosecant

Question 9. For any angle $\theta$ where $\cos \theta \neq 0$, the expression $\frac{\sin \theta}{\cos \theta}$ is equal to____

(A) $\cot \theta$

(B) $\tan \theta$

(C) $\sec \theta$

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

Question 10. In a right triangle with legs of lengths 'a' and 'b' and a hypotenuse of length 'c', the relationship $a^2 + b^2 = c^2$ is known as the____

(A) Law of Sines

(B) Law of Cosines

(C) Pythagorean Theorem

(D) Triangle Inequality

Question 1. The exact value of $\sin 30^\circ$ is____

(A) 0

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

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

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

Question 2. The value of $\tan 45^\circ$ is____

(A) 0

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

(C) 1

(D) $\sqrt{3}$

Question 3. $\cos 60^\circ$ is equal to____

(A) $\sin 30^\circ$

(B) $\cos 30^\circ$

(C) $\sin 60^\circ$

(D) $\tan 30^\circ$

Question 4. The sum $\sin 45^\circ + \cos 45^\circ$ is equal to____

(A) 1

(B) $\sqrt{2}$

(C) 2

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

Question 5. For any angle $\theta$, $\cos (90^\circ - \theta)$ is equivalent to____

(A) $\sin \theta$

(B) $\cos \theta$

(C) $\tan \theta$

(D) $\cot \theta$

Question 6. If A and B are acute angles such that $\sin A = \cos B$, then $A+B$ is equal to____

(A) $45^\circ$

(B) $60^\circ$

(C) $90^\circ$

(D) $180^\circ$

Question 7. The exact value of $\tan 0^\circ$ is equal to____

(A) 0

(B) 1

(C) undefined

(D) $\sqrt{3}$

Question 8. The value of $\cos 90^\circ$ is____

(A) 0

(B) 1

(C) -1

(D) undefined

Question 9. The value of $\tan 90^\circ$ is____

(A) 0

(B) 1

(C) -1

(D) undefined

Question 10. Using complementary angle properties, $\frac{\sin 50^\circ}{\cos 40^\circ}$ simplifies to____

(A) 0

(B) 1

(C) $\tan 50^\circ$

(D) $\cot 40^\circ$

Question 1. The identity $\sin^2 \theta + \cos^2 \theta = 1$ is a fundamental____

(A) Reciprocal identity

(B) Quotient identity

(C) Pythagorean identity

(D) Sum identity

Question 2. For values of $\theta$ where $\tan \theta$ is defined, the expression $\sec^2 \theta - 1$ is equal to____

(A) $\sin^2 \theta$

(B) $\cos^2 \theta$

(C) $\tan^2 \theta$

(D) $\cot^2 \theta$

Question 3. For values of A where $\cot A$ is defined, the expression $1 + \cot^2 A$ is equivalent to____

(A) $\sin^2 A$

(B) $\cos^2 A$

(C) $\sec^2 A$

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

Question 4. The quotient identity for $\cot \theta$ states that $\cot \theta =$____

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

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

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

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

Question 5. The reciprocal identity $\text{cosec } \theta = \frac{1}{\sin \theta}$ is true for all $\theta$ where____

(A) $\sin \theta = 0$

(B) $\sin \theta \neq 0$

(C) $\cos \theta = 0$

(D) $\cos \theta \neq 0$

Question 6. The expression $\sec^2 A (1 - \sin^2 A)$ simplifies to____

(A) $\tan^2 A$

(B) $\cot^2 A$

(C) 1

(D) $\sec^2 A$

Question 7. The expression $\sin \theta \cdot \cot \theta$ simplifies to____

(A) $\sin \theta$

(B) $\cos \theta$

(C) $\tan \theta$

(D) $\sec \theta$

Question 8. The expression $(\sin \theta + \cos \theta)^2 - 1$ simplifies to____

(A) 0

(B) 1

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

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

Question 9. For values of $\theta$ where $\sec \theta$ and $\tan \theta$ are defined, the expression $(\sec \theta - \tan \theta)(\sec \theta + \tan \theta)$ simplifies to____

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

(B) 1

(C) 0

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

Question 10. The expression $\frac{\tan \theta}{\sec \theta}$ simplifies to____

(A) $\sin \theta$

(B) $\cos \theta$

(C) $\tan \theta$

(D) $\cot \theta$

Question 1. One degree is divided into 60 units called____

(A) Radians

(B) Seconds

(C) Minutes

(D) Grads

Question 2. The degree measure of a straight angle is____

(A) $90^\circ$

(B) $180^\circ$

(C) $270^\circ$

(D) $360^\circ$

Question 3. A radian is the angle subtended at the center of a circle by an arc equal in length to the____

(A) Diameter

(B) Radius

(C) Circumference

(D) Chord

Question 4. To convert an angle measured in radians to degrees, you should multiply the radian measure by____

(A) $\pi/180$

(B) $180/\pi$

(C) $\pi/360$

(D) $360/\pi$

Question 5. To convert an angle measured in degrees to radians, you should multiply the degree measure by____

(A) $\pi/180$

(B) $180/\pi$

(C) $\pi/360$

(D) $360/\pi$

Question 6. $150^\circ$ is equivalent to how many radians?

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

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

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

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

Question 7. $\frac{2\pi}{3}$ radians is equivalent to how many degrees?

(A) $120^\circ$

(B) $150^\circ$

(C) $210^\circ$

(D) $240^\circ$

Question 8. The length of an arc $l$ in a circle with radius $r$ and central angle $\theta$ (in radians) is given by $l=$____

(A) $r + \theta$

(B) $r \theta$

(C) $r / \theta$

(D) $\theta / r$

Question 9. The area of a sector of a circle with radius $r$ and central angle $\theta$ (in radians) is given by Area =____

(A) $r \theta^2$

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

(C) $\frac{1}{2} r^2 \theta$

(D) $r^2 \theta$

Question 10. A complete angle ($360^\circ$) is equivalent to____

(A) $\pi$ radians

(B) $2\pi$ radians

(C) $\pi/2$ radians

(D) $4\pi$ radians

Question 1. In the unit circle, for an angle $\theta$ in standard position whose terminal side intersects the circle at point $(x, y)$, $\cos \theta$ is the____

(A) x-coordinate

(B) y-coordinate

(C) ratio $y/x$

(D) distance from origin

Question 2. In the unit circle, for an angle $\theta$ in standard position whose terminal side intersects the circle at point $(x, y)$, $\sin \theta$ is the____

(A) x-coordinate

(B) y-coordinate

(C) ratio $x/y$

(D) angle measure

Question 3. The domain of the function $y = \sin x$ is____

(A) $[-1, 1]$

(B) $[0, 2\pi]$

(C) All real numbers ($\mathbb{R}$)

(D) Integers ($\mathbb{Z}$)

Question 4. The range of the function $y = \cos x$ is____

(A) $[-1, 1]$

(B) $[0, 1]$

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

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

Question 5. In Quadrant I of the coordinate plane, all trigonometric functions are____

(A) Positive

(B) Negative

(C) Zero

(D) Undefined

Question 6. In Quadrant II, the sine function is positive, and the cosine function is____

(A) Positive

(B) Negative

(C) Zero

(D) Positive or Negative depending on the angle

Question 7. In Quadrant III, the tangent function is positive, and the sine function is____

(A) Positive

(B) Negative

(C) Zero

(D) Positive or Negative depending on the angle

Question 8. In Quadrant IV, the cosine function is positive, and the sine function is____

(A) Positive

(B) Negative

(C) Zero

(D) Positive or Negative depending on the angle

Question 9. The principal period of both $\sin x$ and $\cos x$ is____

(A) $\pi$

(B) $2\pi$

(C) $\pi/2$

(D) $4\pi$

Question 10. The principal period of both $\tan x$ and $\cot x$ is____

(A) $\pi$

(B) $2\pi$

(C) $\pi/2$

(D) $4\pi$

Question 1. The graph of $y = \sin x$ passes through the point____

(A) $(0, 1)$

(B) $(1, 0)$

(C) $(0, 0)$

(D) $(\pi/2, 1/2)$

Question 2. The graph of $y = \cos x$ passes through the point____

(A) $(0, 0)$

(B) $(0, 1)$

(C) $(1, 0)$

(D) $(\pi, 0)$

Question 3. The maximum value of both $y = \sin x$ and $y = \cos x$ is____

(A) 0

(B) 1

(C) $\pi$

(D) 2

Question 4. The minimum value of both $y = \sin x$ and $y = \cos x$ is____

(A) 0

(B) 1

(C) -1

(D) $-\infty$

Question 5. The graph of $y = \tan x$ has vertical asymptotes where $\cos x$ is____

(A) 0

(B) 1

(C) -1

(D) undefined

Question 6. The graph of $y = \cot x$ has vertical asymptotes where $\sin x$ is____

(A) 0

(B) 1

(C) -1

(D) undefined

Question 7. In the graph of $y = A \sin(Bx)$, the period is given by____

(A) $|A|$

(B) $|B|$

(C) $2\pi/|B|$

(D) $\pi/|B|$

Question 8. In the graph of $y = A \cos(Bx)$, the amplitude is given by____

(A) $|A|$

(B) $|B|$

(C) $2\pi/|B|$

(D) $\pi/|B|$

Question 9. The graph of $y = \sin x$ is symmetric about the____

(A) x-axis

(B) y-axis

(C) origin

(D) line $y=1$

Question 10. The graph of $y = \cos x$ is symmetric about the____

(A) x-axis

(B) y-axis

(C) origin

(D) line $x=1$

Question 1. The formula for $\sin(A+B)$ is____

(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 2. 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 3. The formula for $\tan(A+B)$ is____

(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 4. $\sin 2A$ is equal to____

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

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

(C) $2 \sin A$

(D) $2 \cos A$

Question 5. $\cos 2A$ is equal to $2\cos^2 A -$____

(A) $\cos^2 A$

(B) $\sin^2 A$

(C) 1

(D) 0

Question 6. $\cos 2A$ is equal to $1 - 2$____$^{2}$A

(A) $\sin$

(B) $\cos$

(C) $\tan$

(D) $\cot$

Question 7. $\tan 2A$ is equal to $\frac{2\tan A}{1 - ____}$

(A) $\tan A$

(B) $\tan^2 A$

(C) $2 \tan A$

(D) $2 \tan^2 A$

Question 8. $\sin 3A$ is equal to $3\sin A - 4$____$^{3}$A

(A) $\cos$

(B) $\sin$

(C) $\tan$

(D) $\cot$

Question 9. $\cos 3A$ is equal to $4\cos^3 A - 3$____$^{3}$A

(A) $\sin$

(B) $\cos$

(C) $\tan$

(D) $\cot$

Question 10. The identity $1 - \cos \theta$ is equal to $2$____$^{2} (\theta/2)$

(A) $\sin$

(B) $\cos$

(C) $\tan$

(D) $\cot$

Question 1. The formula for $2 \sin A \cos B$ is $\sin(A+B) +$____$

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

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

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

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

Question 2. The formula for $2 \cos A \cos B$ is $\cos(A+B) +$____$

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

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

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

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

Question 3. The formula for $2 \sin A \sin B$ is $\cos(A-B) -$____

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

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

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

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

Question 4. The formula for $\sin C + \sin D$ is $2 \sin \left(\frac{C+D}{2}\right) \cos \left(\frac{____}{2}\right)$

(A) $C+D$

(B) $C-D$

(C) $D-C$

(D) $C$

Question 5. The formula for $\cos C + \cos D$ is $2 \cos \left(\frac{C+D}{2}\right) \cos \left(\frac{____}{2}\right)$

(A) $C+D$

(B) $C-D$

(C) $D-C$

(D) $C$

Question 6. The formula for $\sin C - \sin D$ is $2 \cos \left(\frac{C+D}{2}\right) \sin \left(\frac{____}{2}\right)$

(A) $C+D$

(B) $C-D$

(C) $D-C$

(D) $C$

Question 7. The formula for $\cos C - \cos D$ is $-2 \sin \left(\frac{C+D}{2}\right) \sin \left(\frac{____}{2}\right)$

(A) $C+D$

(B) $C-D$

(C) $D-C$

(D) $C$

Question 8. The expression $\frac{\sin x + \sin y}{\cos x + \cos y}$ simplifies to____

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

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

(C) $\tan(x+y)$

(D) $\cot(x+y)$

Question 9. The expression $\frac{\sin x - \sin y}{\cos x + \cos y}$ simplifies to____

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

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

(C) $\tan(x-y)$

(D) $\cot(x-y)$

Question 10. The value of $\sin 75^\circ + \sin 15^\circ$ is____

(A) $2 \sin 45^\circ \cos 30^\circ$

(B) $2 \cos 45^\circ \sin 30^\circ$

(C) $\sin 90^\circ$

(D) $\cos 60^\circ$

Question 1. The principal solution of a trigonometric equation is the solution that lies in the specific range defined for the____

(A) Entire domain of the function

(B) Principal value branch of the corresponding inverse trigonometric function

(C) Interval $[0, 2\pi)$

(D) First quadrant

Question 2. The general solution of $\sin x = \sin \alpha$ is $x =$____

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

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

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

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

Question 3. The general solution of $\cos x = \cos \alpha$ is $x =$____

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

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

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

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

Question 4. The general solution of $\tan x = \tan \alpha$ is $x =$____

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

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

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

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

Question 5. The general solution of $\sin x = 0$ is $x =$____

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

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

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

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

Question 6. The general solution of $\cos x = 0$ is $x =$____

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

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

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

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

Question 7. The general solution of $\tan x = 0$ is $x =$____

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

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

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

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

Question 8. The general solution of $\sin^2 x = \sin^2 \alpha$ is $x =$____

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

(B) $2n\pi \pm \alpha, n \in \mathbb{Z}$

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

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

Question 9. To solve a trigonometric equation like $2\cos^2 x - \cos x - 1 = 0$, we can first treat it as a quadratic in____

(A) $x$

(B) $\sin x$

(C) $\cos x$

(D) $\tan x$

Question 10. For $\sin x = k$, where $-1 \leq k \leq 1$, the principal value $\alpha = \sin^{-1} k$ lies in the interval____

(A) $[0, \pi]$

(B) $[-\pi/2, \pi/2]$

(C) $(0, \pi)$

(D) $(-\pi/2, \pi/2)$

Question 1. The function $\sin^{-1} x$ is the inverse of the sine function with a restricted domain of____

(A) $[-1, 1]$

(B) $\mathbb{R}$

(C) $[-\pi/2, \pi/2]$

(D) $[0, \pi]$

Question 2. The domain of $\cos^{-1} x$ is____

(A) $[0, \pi]$

(B) $[-1, 1]$

(C) $\mathbb{R}$

(D) $[-\pi/2, \pi/2]$

Question 3. The principal value branch (range) of $\sin^{-1} x$ is____

(A) $[0, \pi]$

(B) $[-\pi/2, \pi/2]$

(C) $(-\pi/2, \pi/2)$

(D) $(0, \pi)$

Question 4. The principal value branch (range) of $\cos^{-1} x$ is____

(A) $[0, \pi]$

(B) $[-\pi/2, \pi/2]$

(C) $(-\pi/2, \pi/2)$

(D) $(0, \pi)$

Question 5. The principal value branch (range) of $\tan^{-1} x$ is____

(A) $[0, \pi]$

(B) $[-\pi/2, \pi/2]$

(C) $(-\pi/2, \pi/2)$

(D) $(0, \pi)$

Question 6. For $x \in [-1, 1]$, $\sin^{-1} x + \cos^{-1} x$ is equal to____

(A) 0

(B) $\pi/4$

(C) $\pi/2$

(D) $\pi$

Question 7. For $x \in \mathbb{R}$, $\tan^{-1} x + \cot^{-1} x$ is equal to____

(A) 0

(B) $\pi/4$

(C) $\pi/2$

(D) $\pi$

Question 8. For $|x| \geq 1$, $\sec^{-1} x + \text{cosec}^{-1} x$ is equal to____

(A) 0

(B) $\pi/4$

(C) $\pi/2$

(D) $\pi$

Question 9. For $x \in [-1, 1]$, $\sin^{-1} (-x)$ is equal to____

(A) $\sin^{-1} x$

(B) $-\sin^{-1} x$

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

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

Question 10. For $x \in [-1, 1]$, $\cos^{-1} (-x)$ is equal to____

(A) $\cos^{-1} x$

(B) $-\cos^{-1} x$

(C) $\pi - \cos^{-1} x$

(D) $\pi + \cos^{-1} x$

Question 1. The angle of elevation is formed between the horizontal line and the line of sight when the observer is looking____

(A) Downwards to an object

(B) Upwards to an object

(C) Parallel to the ground

(D) Perpendicular to the ground

Question 2. The angle of depression is formed between the horizontal line and the line of sight when the observer is looking____

(A) Downwards to an object

(B) Upwards to an object

(C) Parallel to the ground

(D) Perpendicular to the ground

Question 3. The angle of elevation from point A to point B is equal to the angle of depression from point B to point A because they are____

(A) Corresponding angles

(B) Alternate interior angles

(C) Vertically opposite angles

(D) Adjacent angles

Question 4. If the angle of elevation of the top of a tower from a point 50m away from its base is $30^\circ$, the height of the tower is given by $50 \times$____

(A) $\sin 30^\circ$

(B) $\cos 30^\circ$

(C) $\tan 30^\circ$

(D) $\cot 30^\circ$

Question 5. If a kite string is 100m long and makes an angle of $45^\circ$ with the ground, the height of the kite is given by $100 \times$____

(A) $\sin 45^\circ$

(B) $\cos 45^\circ$

(C) $\tan 45^\circ$

(D) $\cot 45^\circ$

Question 6. If a ladder is 10m long and makes an angle of $60^\circ$ with the ground, the distance of the foot of the ladder from the wall is given by $10 \times$____

(A) $\sin 60^\circ$

(B) $\cos 60^\circ$

(C) $\tan 60^\circ$

(D) $\sec 60^\circ$

Question 7. Which of the following trigonometric ratios is NOT typically used in basic heights and distances problems involving a single right triangle?

(A) Sine

(B) Cosine

(C) Tangent

(D) Cosecant

Question 8. If two points of observation are on opposite sides of a tower, the distance between the points is the sum of their distances from the____

(A) Top of the tower

(B) Base of the tower

(C) Midpoint of the tower

(D) Observer's eye level

Question 9. Heights and distances problems primarily utilize the properties of____

(A) Oblique triangles

(B) Isosceles triangles

(C) Right-angled triangles

(D) Scalene triangles

Question 10. The line connecting the observer's eye to the object being viewed is called the____

(A) Horizontal line

(B) Vertical line

(C) Line of sight

(D) Base line