CBSE Class 10 Mathematics Introduction to Trigonometry MCQs Set J

Question. \(\Delta ABC\) is right angled at A. If AC = 8 cm and AB = 6 cm, what is the value of \(\text{cosec } B\)?
(a) \(\frac{5}{4}\)
(b) \(\frac{3}{4}\)
(c) \(\frac{4}{3}\)
(d) \(\frac{4}{5}\)
Answer: A

Question. \(\Delta ABC\) is right angled at A. If \(BC = \sqrt{2}\) and AB = AC = 1, what is the measure of \(\angle B\)?
(a) \(60^\circ\)
(b) \(45^\circ\)
(c) \(30^\circ\)
(d) \(90^\circ\)
Answer: B

Question. What is the value of \(\theta\) for which \(\tan \theta = \cot \theta\)?
(a) \(60^\circ\)
(b) \(45^\circ\)
(c) \(90^\circ\)
(d) \(0^\circ\)
Answer: B

Question. Given \(\sin \theta + \frac{1}{\sin \theta} = 4\), what is the value of \(\sin^2 \theta + \frac{1}{\sin^2 \theta}\)?
(a) 20
(b) 16
(c) 14
(d) 4
Answer: C

Question. Given \(\sin^2 \theta + \frac{1}{\sin^2 \theta}\), what is the value of \(\sin \theta + \cos \theta\)?
(a) \(\frac{25}{31}\)
(b) \(\frac{31}{25}\)
(c) \(\frac{24}{25}\)
(d) \(\frac{31}{24}\)
Answer: B

Question. Find the value of \(\cos 30^\circ \cos 45^\circ - \sin 30^\circ \sin 45^\circ\).
(a) \(\frac{\sqrt{6} + 1}{2}\)
(b) \(\frac{\sqrt{6} + \sqrt{2}}{4}\)
(c) \(\frac{\sqrt{6} - \sqrt{2}}{8}\)
(d) \(\frac{\sqrt{2}(\sqrt{3} - 1)}{4}\)
Answer: D

Question. What is the value of \(\tan 7^\circ \tan 23^\circ \tan 60^\circ \tan 67^\circ \tan 83^\circ\)?
(a) \(\frac{1}{\sqrt{3}}\)
(b) \(\sqrt{3}\)
(c) 1
(d) \(\infty\)
Answer: B

Question. \(\Delta ABC\) is an isosceles triangle with the unequal side measuring 12 cm. If both the equal sides measure 19 cm, what is the measure of \(\angle BAC\)?
(a) \(36.8^\circ\)
(b) \(68^\circ\)
(c) \(38^\circ\)
(d) \(60^\circ\)
Answer: A

Question. If \(\sec \theta = 2p\) and \(\tan \theta = \frac{2}{p}\) find the value \(2\left(p^2 - \frac{1}{p^2}\right)\).
(a) \(\frac{1}{2}\)
(b) \(\frac{1}{\sqrt{2}}\)
(c) \(\frac{1}{4}\)
(d) \(\frac{1}{8}\)
Answer: A

Question. What is the value of \(\theta\) if \(\sqrt{3} \tan 2\theta - 3 = 0\)?
(a) \(45^\circ\)
(b) \(90^\circ\)
(c) \(30^\circ\)
(d) \(60^\circ\)
Answer: C

Question. Given \(\sin \theta + \cos \theta = \sqrt{3}\), what is the value of \(\tan \theta + \cot \theta\)?
(a) \(\frac{1}{\sqrt{2}}\)
(b) \(\frac{2}{\sqrt{3}}\)
(c) \(\frac{\sqrt{3}}{2}\)
(d) 1
Answer: D

Question. If \(\frac{x(\tan 26^\circ + \tan 19^\circ)}{(1 - \tan 26^\circ \tan 19^\circ)} = \cos 60^\circ\), what is the value of \(x\)?
(a) 1
(b) \(\sqrt{2}\)
(c) 2
(d) \(\sqrt{3}\)
Answer: C

Question. If \(\sin \left[ 2 \left( \frac{2}{1} \cdot \frac{3}{2} \cdot \frac{4}{3} \cdots \frac{x-1}{x-2} \right) \right] = 1\), \(0 < x < 100\), find the value of \(x\).
(a) \(91^\circ\)
(b) \(80^\circ\)
(c) \(49^\circ\)
(d) \(46^\circ\)
Answer: D

Question. What is the angle between the hour and minute hands of a clock at 02:15 hours?
(a) \(15^\circ\)
(b) \(7 \frac{1}{2}^\circ\)
(c) \(22 \frac{1}{2}^\circ\)
(d) \(30^\circ\)
Answer: C

Question. If \(\tan \theta + \sec \theta = 2\), \(0 \le \theta \le \frac{\pi}{2}\) find the value of \(\tan \theta\).
(a) \(\frac{3}{4}\)
(b) \(\frac{5}{4}\)
(c) \(\frac{3}{2}\)
(d) \(\frac{5}{2}\)
Answer: A

Question. Graphs of \(y = \sin x\) and \(y = \cos x\), where \(0 \le x \le \frac{\pi}{2}\) intersect at a point. Find abscissa.
(a) \(\frac{\pi}{6}\)
(b) \(\frac{\pi}{4}\)
(c) \(\frac{\pi}{3}\)
(d) 0
Answer: B

Question. Find the value of \(\sin 15^\circ\).
(a) \(\frac{\sqrt{3} + 1}{\sqrt{2}}\)
(b) \(\frac{\sqrt{3} - 1}{\sqrt{2}}\)
(c) \(\frac{\sqrt{3} + 1}{2\sqrt{2}}\)
(d) \(\frac{\sqrt{3} - 1}{2\sqrt{2}}\)
Answer: D

Question. What is the value of \(\sin 0^\circ + \cos 30^\circ - \tan 45^\circ + \text{cosec } 60^\circ + \cot 90^\circ\)?
(a) \(\frac{7\sqrt{3} - 6}{6}\)
(b) \(\frac{6 + 7\sqrt{3}}{6}\)
(c) 0
(d) 2
Answer: A

Question. If \(\sin \theta = -\frac{1}{2}\), what are the respective possible values of \(\theta\) between 0 and \(2\pi\)?
(a) \(210^\circ\) and \(300^\circ\)
(b) \(240^\circ\) and \(330^\circ\)
(c) \(240^\circ\) and \(300^\circ\)
(d) \(210^\circ\) and \(330^\circ\)
Answer: D

Question. In terms of radians, what is the equivalent of \(45^\circ\)?
(a) \(25\pi\)
(b) \(0.25\pi\)
(c) \(\frac{180^\circ}{45^\circ} \pi\)
(d) \(\frac{45^\circ}{\pi}\)
Answer: D

Question. Find the value of \(\sqrt{\frac{1 - \cos A}{1 + \cos A}}\)
(a) \(\sec A - \cot A\)
(b) \(\text{cosec } A - \cot A\)
(c) 0
(d) 1
Answer: C

Question. Find the value of \(\frac{5\cos\alpha - 4}{3 - 5\sin\alpha} - \frac{3 + 5\sin\alpha}{4 + 5\cos\alpha}\)
(a) - 1
(b) 5
(c) 1
(d) 0
Answer: D

Question. If \(x = 3 \cos A \cos B\), \(y = 3 \cos A \sin B\) and \(z = 3 \sin A\), find the value of \(x^2 + y^2 + z^2\).
(a) 3
(b) 6
(c) 12
(d) 9
Answer: D

Question. Find the value of \(\frac{1 + \tan 75^\circ}{1 - \tan 75^\circ}\)
(a) \(-\frac{2}{\sqrt{3}}\)
(b) \(\sqrt{3}\)
(c) \(-\sqrt{3}\)
(d) \(\frac{1}{\sqrt{3}}\)
Answer: C

Question. If 8 \(\tan A = 15\), find the value of \(\frac{\sin A - \cos A}{\sin A + \cos A}\)
(a) \(\frac{7}{23}\)
(b) \(\frac{11}{23}\)
(c) \(\frac{13}{23}\)
(d) \(\frac{17}{23}\)
Answer: A

Question. If \(4 \sin \theta = 3 \cos \theta\), find \(\frac{\sec^2 \theta}{4(1 - \tan^2 \theta)}\).
(a) \(\frac{25}{16}\)
(b) \(\frac{25}{28}\)
(c) \(\frac{1}{4}\)
(d) \(\frac{5}{6}\)
Answer: B

Question. Find the value of \(\cos 1^\circ \cos 2^\circ \cos 3^\circ \cdots \cos 89^\circ \cos 90^\circ\).
(a) 1
(b) \(\frac{1}{2}\)
(c) \(\frac{1}{\sqrt{2}}\)
(d) 0
Answer: D

Question. If \(\tan x = \frac{x}{y}\), where x and y are whole numbers, find \(\sin x\).
(a) \(\frac{y}{\sqrt{y^2 - x^2}}\)
(b) \(\frac{x}{\sqrt{x^2 + y^2}}\)
(c) \(\frac{y}{\sqrt{x^2 + y^2}}\)
(d) \(\frac{x}{\sqrt{y^2 - x^2}}\)
Answer: B

Question. Find \(\frac{\sin^3 \phi + \cos^3 \phi}{\sin \phi + \cos \phi}\)
(a) \(1 + \sin \phi \cos \phi\)
(b) \(1 - \sin \phi \cos \phi\)
(c) \(1 - \sin \phi \tan \phi\)
(d) 1
Answer: A

Question. If \(x = a \sec \theta + b \tan \theta\) and \(y = b \sec \theta + a \tan \theta\), find \(x^2 - y^2\).
(a) \(4ab \sec \theta \tan \theta + a\)
(b) \(a^2 - b^2\)
(c) \(b^2 - a^2\)
(d) \(a^2 + b^2\)
Answer: B

Question. If \(\sec \theta + \tan \theta = p\), what is the value of \(\cos \theta\)?
(a) \(\frac{p^2 + 1}{p^2 - 1}\)
(b) \(\frac{p^2 - 1}{(p^2 + 1)^2}\)
(c) \(\frac{2p}{p^2 + 1}\)
(d) \(\frac{4p^2}{(p^2 + 1)^2}\)
Answer: C

Question. What is the numerical value of the expression \(\frac{\sin 9^\circ}{\sin 48^\circ} - \frac{\cos 81^\circ}{\cos 42^\circ}\)?
(a) 1
(b) \(\frac{1}{2}\)
(c) 0
(d) -1
Answer: C

Question. Find the value of the expression \([\text{cosec}(75^\circ + \theta) - \sec(15^\circ - \theta) - \tan(55^\circ + \theta) + \cot(35^\circ - \theta)]\)
(a) - 1
(b) 0
(c) 1
(d) \(\frac{3}{2}\)
Answer: B

Question. If \(\cos(\alpha + \beta) = 0\), what is the value of \(\sin(\alpha - \beta)\)?
(a) \(\cos \beta\)
(b) \(\cos 2\beta\)
(c) \(\sin \alpha\)
(d) \(\sin 2\alpha\)
Answer: B

Question. If \(\cos 9\alpha = \sin \alpha\) and \(9\alpha < 90^\circ\), what is the value of \(\tan 5\alpha\)?
(a) \(\frac{1}{\sqrt{3}}\)
(b) \(\sqrt{3}\)
(c) 1
(d) \(\frac{1}{2}\)
Answer: C

Question. If \(\Delta ABC\) is right angled at C, find the value of \(\cos(A + B)\).
(a) 0
(b) 1
(c) \(\frac{1}{2}\)
(d) \(\frac{\sqrt{3}}{2}\)
Answer: A

Question. If \(\sin A + \sin^2 A = 1\), find the value of the expression \((\cos^2 A + \cos^4 A)\).
(a) 1
(b) \(\frac{1}{2}\)
(c) 2
(d) 3
Answer: A

Question. Given that \(\sin \alpha = \frac{1}{2}\) and \(\cos \beta = \frac{1}{2}\), find the value of \((\alpha + \beta)\).
(a) \(0^\circ\)
(b) \(30^\circ\)
(c) \(60^\circ\)
(d) \(90^\circ\)
Answer: D

Question. Find the value of the expression \(\frac{\sin^2 22^\circ + \sin^2 68^\circ}{\cos^2 22^\circ + \cos^2 68^\circ} + \sin^2 63^\circ + \cos 63^\circ \sin 27^\circ\)
(a) 3
(b) 2
(c) 1
(d) 0
Answer: B

Question. If \(\sin \theta - \cos \theta = 0\), find the value of \((\sin^4 \theta + \cos^4 \theta)\)
(a) 1
(b) \(\frac{3}{4}\)
(c) \(\frac{1}{2}\)
(d) \(\frac{1}{4}\)
Answer: C

Question. Find the value of \(\sin(45^\circ + \theta) - \cos(45^\circ - \theta)\).
(a) \(2\cos \theta\)
(b) 0
(c) \(2\sin \theta\)
(d) 1
Answer: B

Question. What is the value of \(\sin^2 5^\circ + \sin^2 10^\circ + \sin^2 80^\circ + \sin^2 85^\circ\)?
(a) 0
(b) 1
(c) 2
(d) 3
Answer: C

Question. If \(\sin B = \frac{1}{2}\), find the value of \(3 \cos B - 4 \cos^3 B\).
(a) \(\frac{1}{2}\)
(b) 1
(c) 2
(d) 0
Answer: D

Question. If \(\tan x = \sin 45^\circ \cos 45^\circ + \sin 30^\circ\), find the value of \(x\).
(a) \(30^\circ\)
(b) \(60^\circ\)
(c) \(45^\circ\)
(d) \(90^\circ\)
Answer: C

Question. If \(\sin(A+B) = 1\) and \(\cos(A-B) = \frac{\sqrt{3}}{2}\), find A and B.
(a) \(45^\circ, 45^\circ\)
(b) \(90^\circ, 45^\circ\)
(c) \(45^\circ, 30^\circ\)
(d) \(60^\circ, 30^\circ\)
Answer: D

Question. If \(4 \tan \theta = 1\), find the value of \(\frac{4\sin \theta - 2\cos \theta}{4\sin \theta + 3\cos \theta}\)
(a) \(\frac{1}{2}\)
(b) \(\frac{1}{6}\)
(c) \(\frac{2}{3}\)
(d) \(\frac{1}{3}\)
Answer: B

Question. Find the value \(\sin^2 30^\circ \cos^2 45^\circ + 4\tan^2 30^\circ + \frac{1}{2}\sin^2 90^\circ - 2\cos^2 90^\circ + \frac{1}{24}\)
(a) 1
(b) 2
(c) \(\sqrt{2}\)
(d) \(\sqrt{3}\)
Answer: B

Question. Find the value of \(\frac{2}{3}(\cos^4 30^\circ - \sin^4 45^\circ) - 3(\sin^2 60^\circ - \sec^2 45^\circ) + \frac{1}{4}\cot^2 30^\circ\)
(a) \(\frac{15}{4}\)
(b) \(\frac{3}{4}\)
(c) \(2 \frac{65}{4}\)
(d) \(4 \frac{17}{24}\)
Answer: D

Question. If \(\cos 3x = \cos 30^\circ \sin 60^\circ - \sin 30^\circ \cos 60^\circ\), find the value of x.
(a) \(60^\circ\)
(b) \(45^\circ\)
(c) \(20^\circ\)
(d) \(30^\circ\)
Answer: C

Question. If \(\frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}\) find the acute angle \(\theta\).
(a) \(90^\circ\)
(b) \(45^\circ\)
(c) \(30^\circ\)
(d) \(60^\circ\)
Answer: D

Question. If \(2(\cos \theta + \sec \theta) = 5\), what is the value of \(\cos^2 \theta + \sec^2 \theta\)?
(a) \(\frac{25}{2}\)
(b) \(\frac{5}{4}\)
(c) \(\frac{17}{4}\)
(d) \(\frac{4}{17}\)
Answer: D

Question. How is \(\cot \theta\) expressed in terms of \(\sin \theta\)?
(a) \(\frac{1}{\sqrt{1 - \sin^2 \theta}}\)
(b) \(\frac{\sqrt{1 - \sin^2 \theta}}{\sin \theta}\)
(c) \(\frac{1}{\sin \theta}\)
(d) \(\frac{\sin \theta}{\sqrt{1 - \sin^2 \theta}}\)
Answer: B