Trigonometric Functions
Showing all questions for Class 11 Math
Question 1:
Find the radian measures corresponding to the degree measures \(\text{2}{{\text{5}}^{\text{o}}}\)...
Question 2:
Convert 40° 20′ into radian measure....
Question 3:
Convert 6 radians into degree measure...
Question 4:
Find the radius of the circle in which a central angle of 60° intercepts an
arc of length 37.4 cm...
Question 5:
The minute hand of a watch is 1.5 cm long. How far does its tip move in
40 minutes? (Use π = 3.14)....
Question 6:
If the arcs of the same lengths in two circles subtend angles 65°and 110° at the centre, find the ratio of their radii....
Question 7:
A wheel makes 360 revolutions in one minute. Through how many radians does
it turn in one second?...
Question 8:
Find the degree measure of the angle subtended at the centre of a circle of
radius 100 cm by an arc of length 22 cm (use \( \pi =\dfrac{22}{7}\))...
Question 9:
In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of
minor arc of the chord....
Question 10:
If in two circles, arcs of the same length subtend angles 60° and 75° at the
centre, find the ratio of their radii....
Question 11:
Find the angle in radian through which a pendulum swings if its length is 75 cm
and the tip describes an arc of length 10 cm...
Question 12:
If \(\cos x = –\frac{3}{5},\) x lies in the third quadrant, find the values of other five
trigonometric functions....
Question 13:
If \(\cot x = –\frac{5}{12},\) x lies in the second quadrant, find the values of other five
trigonometric functions....
Question 14:
If \(\cot x = \frac{3}{4},\) x lies in the third quadrant, find the values of other five
trigonometric functions....
Question 15:
Find the value of \(\sin\frac{31 \pi}{3}\)
...
Question 16:
Find the value of \(\cos (–1710°)\)...
Question 17:
If \(\cos x = -\frac{1}{2},\) x lies in the third quadrant, find the values of other five
trigonometric functions....
Question 18:
If \(\sin x = \frac{3}{5},\) x lies in the second quadrant, find the values of other five
trigonometric functions....
Question 19:
If \(\sec x = \frac{13}{5},\) x lies in the fourth quadrant, find the values of other five
trigonometric functions....
Question 20:
Prove that:
\(
\sin^2\frac{\pi}{6} + \cos^2\frac{\pi}{3} - \tan^2\frac{\pi}{4} = -\frac{1}{2}.
\)
...
Question 21:
Prove that:
\(
2\sin^2\frac{\pi}{6} + \text{cosec}^2\frac{7\pi}{6}\,\cos^2\frac{\pi}{3} = \frac{3}{2}.
\)...
Question 22:
Prove that:
\(
\cot^2\frac{\pi}{6} + \text{cosec}\frac{5\pi}{6} + 3\tan^2\frac{\pi}{6} = 6.
\)...
Question 23:
Prove that:
\(
2\sin^2\left(\frac{3\pi}{4}\right) + 2\cos^2\left(\frac{\pi}{4}\right) + 2\sec^2\left(\frac{\pi}{3}\right) = 10
\)...
Question 24:
Find the value of:
\(\sin 75^\circ\)...
Question 25:
Find the value of
\(\tan 15^\circ\)...
Question 26:
Prove that:
\(
\cos\left( \frac{\pi}{4} - x \right)\cos\left( \frac{\pi}{4} - y \right) - \sin\left( \frac{\pi}{4} - x \right)\sin\left( \frac{\pi}{4} - y \right) = \sin(x + y)
\)
...
Question 27:
Prove that:
\(
\frac{\tan\left( \frac{\pi}{4} + x \right)}{\tan\left( \frac{\pi}{4} - x \right)} = \left( \frac{1 + \tan x}{1 - \tan x} \right)^2
\)...
Question 28:
Prove that:
\[
\frac{\cos(\pi + x)\cos(-x)}{\sin(\pi - x)\cos\left( \frac{\pi}{2} + x \right)} = \cot^2 x
\]...
Question 29:
Prove that:
\[
\cos\left( \frac{3\pi}{2} + x \right)\cos(2\pi + x)\left[ \cot\left( \frac{3\pi}{2} - x \right) + \cot(2\pi + x) \right] = 1
\]...
Question 30:
Prove that:
\[
\sin(n+1)x \cdot \sin(n+2)x + \cos(n+1)x \cdot \cos(n+2)x = \cos x
\]...
Question 31:
Prove that:
\[
\cos\left( \frac{3\pi}{4} + x \right) - \cos\left( \frac{3\pi}{4} - x \right) = -\sqrt{2} \sin x
\]...
Question 32:
Prove that:
\[
\sin^2(6x) - \sin^2(4x) = \sin(2x) \cdot \sin(10x)
\]...
Question 33:
Prove that:
\[
\cos^2(2x) - \cos^2(6x) = \sin(4x) \cdot \sin(8x)
\]...
Question 34:
Prove that
\[
\sin 2x + 2\sin 4x + \sin 6x = 4\cos^2 x \cdot \sin 4x
\]...
Question 35:
Prove that:
\[
\cot 4x \cdot (\sin 5x + \sin 3x) = \cot x \cdot (\sin 5x - \sin 3x)
\]...
Question 36:
Prove that:
\(
\frac{\sin 5x + \sin 3x}{\cos 5x + \cos 3x} = \tan 4x
\)...
Question 37:
Prove that:
\(
\frac{\sin x - \sin y}{\cos x + \cos y} = \tan\left(\frac{x - y}{2}\right)
\)...
Question 38:
Prove that:
\(
\frac{\sin x + \sin 3x}{\cos x + \cos 3x} = \tan 2x
\)...
Question 39:
Prove that:
\[
\frac{\sin x - \sin 3x}{\sin^2 x - \cos^2 x} = 2 \sin x
\]
...
Question 40:
Prove that:
\(
\frac{\cos 4x + \cos 3x + \cos 2x}{\sin 4x + \sin 3x + \sin 2x} = \cot 3x
\)...
Question 41:
Prove that:
\( \qquad
\cot x \cot 2x - \cot 2x \cot 3x - \cot 3x \cot x = 1
\)
...
Question 42:
Prove that
\[
\tan 4x = \frac{4 \tan x (1 - \tan^2 x)}{1 - 6 \tan^2 x + \tan^4 x}
\]...
Question 43:
Prove that
\(\qquad
\cos 4x = 1 - 8 \sin^2 x \cos^2 x
\)...
Question 44:
Prove that
\( \qquad
\cos 6x = 32 \cos^6 x - 48 \cos^4 x + 18 \cos^2 x - 1
\)
...
Question 45:
Prove that
\[
2\cos\left(\frac{\pi}{13}\right)\cos\left(\frac{9\pi}{13}\right) + \cos\left(\frac{3\pi}{13}\right) + \cos\left(\frac{5\pi}{13}\right) = 0
\]
...
Question 46:
Prove that
\[
(\sin 3x + \sin x)\sin x + (\cos 3x - \cos x)\cos x = 0
\]...
Question 47:
Prove that:
\[
(\cos x + \cos y)^2 + (\sin x - \sin y)^2 = 4\cos^2\left(\frac{x + y}{2}\right)
\]...
Question 48:
Prove that:
\[
(\cos x - \cos y)^2 + (\sin x - \sin y)^2 = 4\sin^2\left(\frac{x - y}{2}\right)
\]...
Question 49:
Prove that \[
\sin x + \sin 3x + \sin 5x + \sin 7x = 4\cos x \cos 2x \sin 4x
\]...
Question 50:
Prove that: \[
\frac{(\sin 7x + \sin 5x) + (\sin 9x + \sin 3x)}{(\cos 7x + \cos 5x) + (\cos 9x + \cos 3x)} = \tan 6x
\]...
Question 51:
Prove that: \[
\sin 3x + \sin 2x - \sin x = 4\sin x \cos\left(\frac{x}{2}\right) \cos\left(\frac{3x}{2}\right)
\]...
Question 52:
Find \(\sin\frac{x}{2}, \cos\frac{x}{2}, \tan\frac{x}{2}\) given \(\tan x = -\frac{4}{3}\), and \(x\) lies in the second quadrant...
Question 53:
Find \(\sin\frac{x}{2}, \cos\frac{x}{2}, \tan\frac{x}{2}\) given \(\cos x = -\frac{1}{3}\), and \(x\) lies in the third quadrant...
Question 54:
Find \(\sin\frac{x}{2}, \cos\frac{x}{2}, \tan\frac{x}{2}\) given \(\sin x = \frac{1}{4}\), and \(x\) lies in the second quadrant...