Some Trig Practice questions           Back to Math 160       Back to Math 161

  1. Trigonometric Functions and Identities

  2. Unit Circle and Trig Values

  3. Find the value of the following trig functions.   solution

    4. (a) \( \cot 0 \)   (b) \( \sec \frac{\pi}{4} \)   (c) \( \csc \frac{\pi}{6} \)   (d) \( \sec \frac{\pi}{3} - \cot \frac{\pi}{4} \)   (e) \( \cos \frac{7\pi}{6}-\frac{1}{2}\cot \frac{5\pi}{6} \)

  4. Find the exact value of each of the following trig functions.   solution

    5. (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) \( \csc \frac{17\pi}{3} \)

  5. Find the exact value of each of the following expressions.  solution

    6. (a) \(\tan 40^{\circ} \cdot \dfrac{\csc 40^{\circ}}{\cos 40^{\circ}} \cdot \dfrac{1}{\sec^2{40}}\)   (b) \({\sin \dfrac{\pi}{5} + \cos \dfrac{\pi}{5}}\: \tan \left(-\dfrac{\pi}{5}\right)\)

  6. Find the exact value of each of the following expressions.  solution

    7. (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) \(\tan 10^\circ \cos 10^\circ \csc 10^\circ \)

  7. Evaluate each expression without using a calculator.   solution

    8.  (a) \(2\sec (\pi/3)-\tan (3\pi/4) \)     (b) \(2\tan (\pi/4) - \dfrac{1}{2\tan(-5\pi/4)} +\sin (5\pi/6) \)

  8. Evaluate each expression without using a calculator.  solution

    9. (a) \(\tan ( -7\pi/4) \cdot \cos (4\pi/3) - \sin (5\pi/6)\cdot \sin (3\pi/2) \)   (b) \( \sin (7\pi/6) + \cos (-\pi/3)-\csc^2(\pi/2) \)

  9. Simplify the trigonometric expressions.  solution

    10. (a) \(\cos \theta \cdot \csc \theta \cdot \tan \theta \)   (b) \(\sec\theta - \sec\theta \sin^2\theta \)   (c) \( (\sin \theta + \cos \theta)^2 + (\sin \theta - \cos \theta)^2\)

  10. Simplify the trigonometric expressions. solution

    11. (a) \(\dfrac{\sin \theta-\sin^3 \theta}{\cos^2 \theta}\)   (b) \( 1 -\dfrac{\sin^2\theta}{1+\cos\theta} \)  

  11. Simplify the trigonometric expressions.  solution

    12. (a) \(\tan\theta + \dfrac{\cos\theta}{1+\sin\theta}\)   (b) \(\dfrac{1-\cos\theta}{\sin\theta}+\dfrac{\sin\theta}{1-\cos\theta}\)

  12. Find the exact value of each expression in radians.   solution

    13. (a) \( \sin^{-1} (1)\)     (b) \(\sin^{-1} \left(\dfrac{1}{2}\right)\)    (c) \(\cos^{-1} (0) \)     (d) \( \tan^{-1} (1)\)   (e) \(\cos^{-1} (-1)\)

  13. Find the exact value of each expression in radians.  solution

    14. (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) \(\cos^{-1} \left(\dfrac{\sqrt{3}}{2}\right)\)

  14. Find the exact value of each expression in radians.  solution

    15. (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) \(\tan^{-1} \left(\tan \dfrac{\pi}{7} \right)\)

  15. Solve each equation on the interval \( 0 \leq \theta < 2\pi\).   solution

    16. (a) \(1-\cos \theta = \frac{1}{2} \)   (b) \( 2\sin \theta + \sqrt 3 = 0 \)   (c) \(2 \cos \theta - \sqrt {2} = 0 \)   (d) \(\sqrt{3} \tan \theta + 3 = 0 \)

  16. Solve each equation on the interval \( 0 \leq \theta < 2\pi\).   solution

    17. (a) \(4\cos^2 \theta - 3 = 0 \)  (b) \( 2\sin^2 \theta - 1 = \sin \theta\)