Find the radian measures corresponding to the following degree measures:
(i) \(25^\circ\) (ii) \(-47^\circ 30'\) (iii) \(240^\circ\) (iv) \(520^\circ\)
(i) \(\dfrac{5\pi}{36}\)
(ii) \(-\dfrac{19\pi}{72}\)
(iii) \(\dfrac{4\pi}{3}\)
(iv) \(\dfrac{26\pi}{9}\)
Find the degree measures corresponding to the following radian measures (Use \(\pi = \dfrac{22}{7}\)):
(i) \(\dfrac{11}{16}\) (ii) \(-4\) (iii) \(\dfrac{5\pi}{3}\) (iv) \(\dfrac{7\pi}{6}\)
(i) \(39^\circ\ 22'\ 30''\)
(ii) \(-229^\circ\ 5'\ 27''\)
(iii) \(300^\circ\)
(iv) \(210^\circ\)
A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?
\(12\pi\)
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 \(\pi = \dfrac{22}{7}\)).
\(12^\circ\ 36'\)
In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of the minor arc of the chord.
\(\dfrac{20\pi}{3}\)
If in two circles, arcs of the same length subtend angles \(60^\circ\) and \(75^\circ\) at the centre, find the ratio of their radii.
\(5 : 4\)
Find the angle in radian through which a pendulum swings if its length is 75 cm and the tip describes an arc of length:
(i) 10 cm (ii) 15 cm (iii) 21 cm
(i) \(\dfrac{2}{15}\)
(ii) \(\dfrac{1}{5}\)
(iii) \(\dfrac{7}{25}\)
cos x = -\( \dfrac{1}{2} \), x lies in the third quadrant. Find the other five trigonometric ratios.
\( \sin x = -\dfrac{\sqrt{3}}{2} \), \( \csc x = -\dfrac{2}{\sqrt{3}} \), \( \sec x = -2 \), \( \tan x = \sqrt{3} \), \( \cot x = \dfrac{1}{\sqrt{3}} \)
sin x = \( \dfrac{3}{5} \), x lies in second quadrant. Find the other five trigonometric ratios.
\( \csc x = \dfrac{5}{3} \), \( \cos x = -\dfrac{4}{5} \), \( \sec x = -\dfrac{5}{4} \), \( \tan x = -\dfrac{3}{4} \), \( \cot x = -\dfrac{4}{3} \)
cot x = \( \dfrac{3}{4} \), x lies in third quadrant. Find the other five trigonometric ratios.
\( \sin x = -\dfrac{4}{5} \), \( \csc x = -\dfrac{5}{4} \), \( \cos x = -\dfrac{3}{5} \), \( \sec x = -\dfrac{5}{3} \), \( \tan x = \dfrac{4}{3} \)
sec x = \( \dfrac{13}{5} \), x lies in fourth quadrant. Find the other five trigonometric ratios.
\( \sin x = -\dfrac{12}{13} \), \( \csc x = -\dfrac{13}{12} \), \( \cos x = \dfrac{5}{13} \), \( \tan x = -\dfrac{12}{5} \), \( \cot x = -\dfrac{5}{12} \)
tan x = -\( \dfrac{5}{12} \), x lies in second quadrant. Find the other five trigonometric ratios.
\( \sin x = \dfrac{5}{13} \), \( \csc x = \dfrac{13}{5} \), \( \cos x = -\dfrac{12}{13} \), \( \sec x = -\dfrac{13}{12} \), \( \cot x = -\dfrac{12}{5} \)
Find \( \sin 765^\circ \).
\( \dfrac{1}{\sqrt{2}} \)
Find \( \csc(-1410^\circ) \).
2
Find \( \tan \left( \dfrac{19\pi}{3} \right) \).
\( \sqrt{3} \)
Find \( \sin\left( -\dfrac{11\pi}{3} \right) \).
\( \dfrac{\sqrt{3}}{2} \)
Find \( \cot\left( -\dfrac{15\pi}{4} \right) \).
1
Prove that \( \sin^2\left(\dfrac{\pi}{6}\right) + \cos^2\left(\dfrac{\pi}{3}\right) - \tan^2\left(\dfrac{\pi}{4}\right) = -\tfrac{1}{2} \).
Prove that \( 2\sin^2\left(\dfrac{\pi}{6}\right) + \csc^2\left(\dfrac{7\pi}{6}\right) \cos^2\left(\dfrac{\pi}{3}\right) = \tfrac{3}{2} \).
Prove that \( \cot^2\left(\dfrac{\pi}{6}\right) + \csc\left(\dfrac{5\pi}{6}\right) + 3\tan^2\left(\dfrac{\pi}{6}\right) = 6 \).
Prove that \( 2\sin^2\left(\dfrac{3\pi}{4}\right) + 2\cos^2\left(\dfrac{\pi}{4}\right) + 2\sec^2\left(\dfrac{\pi}{3}\right) = 10 \).
Find the value of:
(i) \( \sin 75^{\circ} \)
(ii) \( \tan 15^{\circ} \)
(i) \( \sin 75^{\circ} = \dfrac{\sqrt{3}+1}{2\sqrt{2}} \)
(ii) \( \tan 15^{\circ} = 2 - \sqrt{3} \)
Prove that \( \cos\left(\dfrac{\pi}{4} - x\right) \cos\left(\dfrac{\pi}{4} - y\right) - \sin\left(\dfrac{\pi}{4} - x\right) \sin\left(\dfrac{\pi}{4} - y\right) = \sin(x + y) \).
Prove that \( \dfrac{\tan\left(\dfrac{\pi}{4} + x\right)}{\tan\left(\dfrac{\pi}{4} - x\right)} = \left( \dfrac{1 + \tan x}{1 - \tan x} \right)^2 \).
Prove that \( \dfrac{\cos(\pi + x) \cos(-x)}{\sin(\pi - x) \cos\left(\dfrac{\pi}{2} + x\right)} = \cot^2 x \).
Prove that \( \cos\left(\dfrac{3\pi}{2} + x\right) \cos(2\pi + x) \left[ \cot\left(\dfrac{3\pi}{2} - x\right) + \cot(2\pi + x) \right] = 1 \).
Prove that \( \sin(n + 1)x \sin(n + 2)x + \cos(n + 1)x \cos(n + 2)x = \cos x \).
Prove that \( \cos\left(\dfrac{3\pi}{4} + x\right) - \cos\left(\dfrac{3\pi}{4} - x\right) = -\sqrt{2} \sin x \).
Prove that \( \sin^2 6x - \sin^2 4x = \sin 2x \sin 10x \).
Prove that \( \cos^2 2x - \cos^2 6x = \sin 4x \sin 8x \).
Prove that \( \sin 2x + 2 \sin 4x + \sin 6x = 4 \cos^2 x \sin 4x \).
Prove that \( \cot 4x \left( \sin 5x + \sin 3x \right) = \cot x \left( \sin 5x - \sin 3x \right) \).
Prove that \( \dfrac{\cos 9x - \cos 5x}{\sin 17x - \sin 3x} = -\dfrac{\sin 2x}{\cos 10x} \).
Prove that \( \dfrac{\sin 5x + \sin 3x}{\cos 5x + \cos 3x} = \tan 4x \).
Prove that \( \dfrac{\sin x - \sin y}{\cos x + \cos y} = \tan \dfrac{x - y}{2} \).
Prove that \( \dfrac{\sin x + \sin 3x}{\cos x + \cos 3x} = \tan 2x \).
Prove that \( \dfrac{\sin x - \sin 3x}{\sin^2 x - \cos^2 x} = 2 \sin x \).
Prove that \( \dfrac{\cos 4x + \cos 3x + \cos 2x}{\sin 4x + \sin 3x + \sin 2x} = \cot 3x \).
Prove that \( \cot x \cot 2x - \cot 2x \cot 3x - \cot 3x \cot x = 1 \).
Prove that \( \tan 4x = \dfrac{4 \tan x \left(1 - \tan^2 x\right)}{1 - 6 \tan^2 x + \tan^4 x} \).
Prove that \( \cos 4x = 1 - 8 \sin^2 x \cos^2 x \).
Prove that \( \cos 6x = 32 \cos^6 x - 48 \cos^4 x + 18 \cos^2 x - 1 \).