Trigonometry Practice

1. Trigonometric Functions and Identities

2. Unit Circle and Trig Values

Find the value of the following trig functions:
(a) \( \cot 0 \)   (b) \( \sec \frac{\pi}{4} \)   (c) \( \csc \frac{\pi}{6} \)   (d) \( \sec \frac{\pi}{3} - \cot \frac{\pi}{4} \)   (e) \( \cos \frac{7\pi}{6}-\frac{1}{2}\cot \frac{5\pi}{6} \)

Find the exact value of each of the following trig functions:
(a) \( \tan \frac{9\pi}{4} \)   (b) \( \cos \frac{19\pi}{6} \)   (c) \( \sin \frac{11\pi}{4} \)   (d) \( \sec \frac{10\pi}{3} \)   (e) \( \csc \frac{17\pi}{3} \)

Find the exact value of each of the following expressions:
(a) \(\tan 40^{\circ} \cdot \dfrac{\csc 40^{\circ}}{\cos 40^{\circ}} \cdot \dfrac{1}{\sec^2{40}}\)

(b) \({\sin \dfrac{\pi}{5} + \cos \dfrac{\pi}{5}}\: \tan \left(-\dfrac{\pi}{5}\right)\)

Find the exact value of each of the following expressions:
(a) \(\tan 10^\circ -\dfrac{\sin 10^\circ}{\cos 10^\circ}\)  (b) \(\cos^2 \frac{\pi}{5} +\dfrac{1}{\csc^2 \frac{\pi}{5}}\)

(c) \(\sec^2 40^\circ - \tan^2 40^\circ\)  (d) \(\tan 10^\circ \cos 10^\circ \csc 10^\circ \)

Evaluate each expression without using a calculator:
(a) \(2\sec (\pi/3)-\tan (3\pi/4) \)
(b) \(2\tan (\pi/4) - \dfrac{1}{2\tan(-5\pi/4)} +\sin (5\pi/6) \)

Evaluate each expression without using a calculator:
(a) \(\tan ( -7\pi/4) \cdot \cos (4\pi/3) - \sin (5\pi/6)\cdot \sin (3\pi/2) \)
(b) \( \sin (7\pi/6) + \cos (-\pi/3)-\csc^2(\pi/2) \)

Simplify the trigonometric expressions:
(a) \(\cos \theta \cdot \csc \theta \cdot \tan \theta \)
(b) \(\sec\theta - \sec\theta \sin^2\theta \)
(c) \( (\sin \theta + \cos \theta)^2 + (\sin \theta - \cos \theta)^2\)

Simplify the trigonometric expressions:
(a) \(\dfrac{\sin \theta-\sin^3 \theta}{\cos^2 \theta}\)   (b) \( 1 -\dfrac{\sin^2\theta}{1+\cos\theta} \)

Simplify the trigonometric expressions:
(a) \(\tan\theta + \dfrac{\cos\theta}{1+\sin\theta}\)   (b) \(\dfrac{1-\cos\theta}{\sin\theta}+\dfrac{\sin\theta}{1-\cos\theta}\)

Find the exact value of each expression in radians:
(a) \( \sin^{-1} (1)\)   (b) \(\sin^{-1} \left(\dfrac{1}{2}\right)\)   (c) \(\cos^{-1} (0) \)   (d) \( \tan^{-1} (1)\)   (e) \(\cos^{-1} (-1)\)

Find the exact value of each expression in radians:
(a) \(\tan^{-1} \left(-\sqrt{3}\right)\)   (b) \(\sin^{-1} \left(-\dfrac{1}{2} \right)\)   (c) \(\tan^{-1} \left( \dfrac{\sqrt 3}{3} \right)\)   (d) \(\cos^{-1} \left(\dfrac{\sqrt{3}}{2}\right)\)

Find the exact value of each expression in radians:
(a) \(\sin^{-1} \left(\sin \dfrac{\pi}{5}\right)\)   (b) \(\cos^{-1} \left( \cos \dfrac{5\pi}{6} \right)\)   (c) \( \sin^{-1} \left( \ tan \dfrac{3\pi}{4} \right)\)   (d) \(\tan^{-1} \left(\tan \dfrac{\pi}{7} \right)\)

Solve each equation on the interval \( 0 \leq \theta < 2\pi\):
(a) \(1-\cos \theta = \frac{1}{2} \)   (b) \( 2\sin \theta + \sqrt 3 = 0 \)   (c) \(2 \cos \theta - \sqrt {2} = 0 \)   (d) \(\sqrt{3} \tan \theta + 3 = 0 \)

Solve each equation on the interval \( 0 \leq \theta < 2\pi\):
(a) \(4\cos^2 \theta - 3 = 0 \)  (b) \( 2\sin^2 \theta - 1 = \sin \theta\)