Matching Items MCQs for Sub-Topics of Topic 8: Trigonometry

(i) Sine

(ii) Cosine

(iii) Tangent

(iv) Cotangent

(a) $\frac{\text{Adjacent side}}{\text{Opposite side}}$

(b) $\frac{\text{Opposite side}}{\text{Hypotenuse}}$

(c) $\frac{\text{Adjacent side}}{\text{Hypotenuse}}$

(d) $\frac{\text{Opposite side}}{\text{Adjacent side}}$

(i) $\sin \theta$

(ii) $\cos \theta$

(iii) $\tan \theta$

(iv) $\sec \theta$

(a) $\cot \theta$

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

(c) $\tan \theta$

(d) $\sec \theta$

(i) Side opposite to angle A

(ii) Side adjacent to angle A

(iii) Side opposite to angle B

(iv) Side opposite to angle C

(a) BC

(b) AC (Hypotenuse)

(c) AB

(d) Cannot be determined without more information

(i) $\sin 30^\circ$

(ii) $\cos 45^\circ$

(iii) $\tan 60^\circ$

(iv) $\sin 90^\circ$

(a) 1

(b) $\approx 0.707$

(c) 0.5

(d) $\approx 1.732$

(i) $\tan \theta$

(ii) $\cot \theta$

(iii) $\sec \theta$

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

(a) $\frac{1}{\cos \theta}$

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

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

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

(i) $0^\circ$

(ii) $30^\circ$

(iii) $45^\circ$

(iv) $60^\circ$

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

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

(c) $0$

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

(i) $0^\circ$

(ii) $30^\circ$

(iii) $45^\circ$

(iv) $60^\circ$

(a) $\sqrt{3}$

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

(c) 1

(d) 0

(i) $\sin (90^\circ - A)$

(ii) $\cos (90^\circ - A)$

(iii) $\tan (90^\circ - A)$

(iv) $\sec (90^\circ - A)$

(a) $\text{cosec } A$

(b) $\cos A$

(c) $\sin A$

(d) $\cot A$

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

(ii) $2 \tan 45^\circ$

(iii) $\sin^2 45^\circ + \cos^2 45^\circ$

(iv) $\sec 60^\circ$

(a) 1

(b) 2

(c) 1

(d) 2

(i) $\sin^2 25^\circ$

(ii) $\tan 10^\circ$

(iii) $\sec 40^\circ$

(iv) $\sin 10^\circ$

(a) $\cos 80^\circ$

(b) $\cos^2 65^\circ$

(c) $\text{cosec } 50^\circ$

(d) $\cot 80^\circ$

(i) $0^\circ$

(ii) $30^\circ$

(iii) $45^\circ$

(iv) $60^\circ$

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

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

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

(d) $1$

(i) $\sin^2 \theta + \cos^2 \theta$

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

(iii) $\text{cosec}^2 \theta - \cot^2 \theta$

(iv) $1 + \tan^2 \theta$

(a) 1

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

(c) 1

(d) $\sec^2 \theta$

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

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

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

(iv) $\frac{1}{\cos \theta}$

(a) $\cot \theta$

(b) $\sec \theta$

(c) $\tan \theta$

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

(i) $\sin \theta \cdot \text{cosec } \theta$

(ii) $\cos \theta \cdot \tan \theta$

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

(iv) $\sec^2 \theta - 1$

(a) $\tan^2 \theta$

(b) $\cos^2 \theta$

(c) $\sin \theta$

(d) 1

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

(ii) $\frac{1}{\sec \theta - \tan \theta}$

(iii) $\sin^4 \theta + \cos^4 \theta$

(iv) $\frac{1 + \tan^2 \theta}{1 + \cot^2 \theta}$

(a) $1 - 2 \sin^2 \theta \cos^2 \theta$

(b) $\sec \theta + \tan \theta$

(c) $\tan^2 A$

(d) $\tan^2 \theta$

(i) $\sin^2 \theta \cot^2 \theta$

(ii) $\cos^2 \theta \tan^2 \theta$

(iii) $(\sec \theta + \tan \theta)(\sec \theta - \tan \theta)$

(iv) $(\text{cosec } \theta + \cot \theta)(\text{cosec } \theta - \cot \theta)$

(a) $\sin^2 \theta$

(b) 1

(c) 1

(d) $\cos^2 \theta$

(i) $30^\circ$

(ii) $45^\circ$

(iii) $60^\circ$

(iv) $90^\circ$

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

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

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

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

(i) $\pi$

(ii) $2\pi$

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

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

(a) $360^\circ$

(b) $135^\circ$

(c) $150^\circ$

(d) $180^\circ$

(i) Arc Length ($l$)

(ii) Area of Sector ($A$)

(iii) Central Angle ($\theta$ in radians)

(iv) Radius ($r$)

(a) $\frac{l}{r}$

(b) $r\theta$

(c) $\frac{2A}{l}$

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

(i) $1^\circ$ to radians

(ii) 1 radian to degrees

(iii) $180^\circ$ to radians

(iv) $\pi$ radians to degrees

(a) $\pi/180$

(b) $180/\pi$

(c) $180^\circ$

(d) $\pi$

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

(ii) $\pi$ radians

(iii) $60^\circ$

(iv) $120^\circ$

(a) 22 cm

(b) 11 cm

(c) $\frac{44}{3}$ cm

(d) 44 cm

(i) $0$ radians

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

(iii) $\pi$ radians

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

(a) $(0, 1)$

(b) $(-1, 0)$

(c) $(1, 0)$

(d) $(0, -1)$

(i) $0 < \theta < \frac{\pi}{2}$

(ii) $\frac{\pi}{2} < \theta < \pi$

(iii) $\pi < \theta < \frac{3\pi}{2}$

(iv) $\frac{3\pi}{2} < \theta < 2\pi$

(a) Quadrant IV

(b) Quadrant II

(c) Quadrant I

(d) Quadrant III

(i) $\sin x$

(ii) $\cos x$

(iii) $\tan x$

(iv) $\cot x$

(a) $\pi$

(b) $2\pi$

(c) $2\pi$

(d) $\pi$

(i) $\sin \pi$

(ii) $\cos \frac{\pi}{2}$

(iii) $\tan \pi$

(iv) $\sin \frac{3\pi}{2}$

(a) 0

(b) -1

(c) 0

(d) 0

(i) Quadrant II

(ii) Quadrant III

(iii) Quadrant IV

(iv) Quadrant I

(a) $\cos \theta > 0, \sin \theta < 0$

(b) $\sin \theta > 0, \cos \theta < 0$

(c) $\sin \theta > 0, \cos \theta > 0$

(d) $\sin \theta < 0, \cos \theta < 0$

(i) $\sin x$

(ii) $\tan x$

(iii) $\sec x$

(iv) $\cot x$

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

(b) $\mathbb{R}$

(c) $[-1, 1]$

(d) $\mathbb{R}$

(i) $y = \sin x$

(ii) $y = \cos x$

(iii) $y = \tan x$

(iv) $y = \cot x$

(a) $(\frac{\pi}{4}, 1)$

(b) $(0, 0)$

(c) $(\frac{\pi}{2}, 0)$

(d) $(\frac{\pi}{2}, 0)$

(i) $y = \sin x$

(ii) $y = 3 \cos x$

(iii) $y = -2 \sin x$

(iv) $y = \tan x$

(a) Not applicable

(b) 1

(c) 3

(d) 2

(i) $y = \cos x$

(ii) $y = \tan x$

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

(iv) $y = \cos (x/2)$

(a) $4\pi$

(b) $\pi$

(c) $2\pi$

(d) $\pi$

(i) $y = \tan x$

(ii) $y = \cot x$

(iii) $y = \sec x$

(iv) $y = \text{cosec } x$

(a) $x = n\pi$

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

(c) $x = n\pi$

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

(i) $y = 2 \sin x$

(ii) $y = - \cos x$

(iii) $y = 1 + \sin x$

(iv) $y = \sec x$

(a) $[-1, 1]$

(b) $[1, 3]$

(c) $[-2, 2]$

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

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

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

(iii) $\tan(A+B)$

(iv) $\sin 2A$

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

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

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

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

(i) $\cos 2A$

(ii) $\cos 2A$

(iii) $\cos 2A$

(iv) $\tan 2A$

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

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

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

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

(i) $\sin 75^\circ$

(ii) $\cos 15^\circ$

(iii) $\tan 15^\circ$

(iv) $\cos 75^\circ$

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

(b) $2 - \sqrt{3}$

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

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

(i) $1 + \cos 2\theta$

(ii) $1 - \cos 2\theta$

(iii) $\frac{1 + \cos 2\theta}{2}$

(iv) $\frac{1 - \cos 2\theta}{2}$

(a) $\sin^2 \theta$

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

(c) $\cos^2 \theta$

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

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

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

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

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

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

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

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

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

(i) $\sin 3A$

(ii) $\cos 3A$

(iii) $\tan 3A$

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

(a) $\frac{3 \tan A - \tan^3 A}{1 - 3 \tan^2 A}$

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

(c) $\cos 2A$

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

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

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

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

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

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

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

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

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

(i) $\sin C + \sin D$

(ii) $\sin C - \sin D$

(iii) $\cos C + \cos D$

(iv) $\cos C - \cos D$

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

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

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

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

(i) $\sin 5\theta + \sin 3\theta$

(ii) $\cos 4x - \cos 2x$

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

(iv) $2 \cos 3\theta \cos \theta$

(a) $\sin 5A + \sin 3A$

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

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

(d) $-2 \sin 3x \sin x$

(i) $\frac{\sin x + \sin y}{\cos x + \cos y}$

(ii) $\frac{\sin x - \sin y}{\cos x - \cos y}$

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

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

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

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

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

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

(i) $\sin 3\theta \cos \theta$

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

(iii) $\sin 5x \sin x$

(iv) $\cos 2x \sin x$

(a) $\frac{1}{2}(\cos 3x - \cos 5x)$

(b) $\frac{1}{2}(\sin 4\theta + \sin 2\theta)$

(c) $\frac{1}{2}(\cos 6\theta + \cos 2\theta)$

(d) $\frac{1}{2}(\sin 3x - \sin x)$

(i) $\sin x = 1$

(ii) $\cos x = 0$

(iii) $\tan x = \sqrt{3}$

(iv) $\cos x = -1$

(a) $\pi$

(b) $\frac{\pi}{2}, \frac{3\pi}{2}$

(c) $\frac{\pi}{3}, \frac{4\pi}{3}$

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

(i) $\sin x = \sin \alpha$

(ii) $\cos x = \cos \alpha$

(iii) $\tan x = \tan \alpha$

(iv) $\sin^2 x = \sin^2 \alpha$

(a) $x = n\pi \pm \alpha$

(b) $x = n\pi + \alpha$(c) $x = 2n\pi \pm \alpha$

(d) $x = n\pi + (-1)^n \alpha$

(i) $\sin x = 0$

(ii) $\cos x = 1$

(iii) $\tan x = 1$

(iv) $\sin x = -1$

(a) $2\pi$

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

(c) $3\pi$

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

(i) Quadratic in $\sin x$ or $\cos x$

(ii) Involving different angles (e.g., $\sin 2x = \sin x$)

(iii) Simple equation $\sin x = k$

(iv) Involving sum/difference of terms (e.g., $\sin 3x + \sin x = 0$)

(a) Use sum-to-product formulas

(b) Use double/multiple angle identities

(c) Factorization or quadratic formula

(d) Find principal solution and use general solution formula

(i) $x = \pi/6$

(ii) $x = \pi/3$

(iii) $x = \pi/4$

(iv) $x = 0$

(a) $\tan x = 1$

(b) $\cos x = 1$

(c) $\sin x = 1/2$

(d) $\cos x = 1/2$

(i) $\sin x = 1/2$

(ii) $\cos x = \sqrt{3}/2$

(iii) $\tan x = 1/\sqrt{3}$

(iv) $\sin x = -1/\sqrt{2}$

(a) $x = n\pi + (-1)^n (-\pi/4)$

(b) $x = n\pi + \pi/6$

(c) $x = 2n\pi \pm \pi/6$

(d) $x = n\pi + (-1)^n \pi/6$

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

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

(iii) $\tan^{-1} x$

(iv) $\sec^{-1} x$

(a) $\mathbb{R}$

(b) $[-1, 1]$

(c) $[-1, 1]$

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

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

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

(iii) $\tan^{-1} x$

(iv) $\text{cosec}^{-1} x$

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

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

(c) $[0, \pi]$

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

(i) $\sin^{-1} (\frac{\sqrt{3}}{2})$

(ii) $\cos^{-1} (-\frac{1}{2})$

(iii) $\tan^{-1} (-1)$

(iv) $\sin^{-1} (\sin \frac{2\pi}{3})$

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

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

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

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

(i) $\sin^{-1} x + \cos^{-1} x$

(ii) $\tan^{-1} x + \cot^{-1} x$

(iii) $\sec^{-1} x + \text{cosec}^{-1} x$

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

(a) $\pi/2$

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

(c) $\pi/2$

(d) $\pi/2$

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

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

(iii) $\tan (\tan^{-1} x)$

(iv) $\sin (\cos^{-1} x)$

(a) $x$

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

(c) $x$

(d) $x$

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

(ii) $\sin^{-1} (\cos 0)$

(iii) $\cos^{-1} (\sin \frac{\pi}{2})$

(iv) $\sin^{-1} (1) + \cos^{-1} (1)$

(a) $\pi/2$

(b) $\pi/4$

(c) $\pi/2$

(d) $\pi/2$

(i) Angle of Elevation

(ii) Angle of Depression

(iii) Line of Sight

(iv) Horizontal Level

(a) The line from the observer's eye to the object.

(b) The angle between the horizontal line and the line of sight when looking downwards.

(c) The angle between the horizontal line and the line of sight when looking upwards.

(d) A line parallel to the ground passing through the observer's eye.

(i) Given angle of elevation $\theta$ and adjacent distance x, find h.

(ii) Given angle of elevation $\theta$ and hypotenuse (line of sight) L, find h.

(iii) Given angle of depression $\phi$ and horizontal distance x, find height difference h.

(iv) Given angle of elevation $\theta$ and opposite side h, find x.

(a) $\tan \phi = h/x$

(b) $\tan \theta = h/x$

(c) $\sin \theta = h/L$

(d) $\tan \theta = h/x$

(i) Height of object (using tangent)

(ii) Hypotenuse (using cosine)

(iii) Hypotenuse (using sine)

(iv) Adjacent side (using cotangent)

(a) $60\ \text{m}$ (if opposite = 30)

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

(c) $20\sqrt{3}\ \text{m}$ (if adjacent = 30)

(d) $30\sqrt{3}\ \text{m}$ (if opposite = 30)

(i) Distance from foot $30\ \text{m}$, angle of elevation $30^\circ$

(ii) Distance from foot $10\ \text{m}$, angle of elevation $60^\circ$

(iii) Shadow length $50\sqrt{3}\ \text{m}$, sun's altitude $30^\circ$

(iv) Angle of elevation changes from $30^\circ$ to $45^\circ$ as observer moves $100\ \text{m}$ towards tower

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

(b) $50\ \text{m}$

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

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

(i) Tower and building problem (angles of elevation/depression from building top)

(ii) Two points on ground, same line as base (different angles of elevation)

(iii) Cloud and its reflection in water (angles of elevation/depression from a point above water)

(iv) Tree broken by wind

(a) Use trigonometric ratios in two right triangles and relate them using horizontal distance.

(b) Sum of the broken part (hypotenuse) and the standing part (adjacent/opposite).

(c) Use trigonometric ratios in two right triangles and relate them using the height difference and equal horizontal distance.

(d) Use trigonometry to find the height of the cloud and the depth of the reflection.