Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

5.4E: The Other Trigonometric Functions (Exercises)

  • Page ID
    13915
  • [ "article:topic", "license:ccbysa", "showtoc:no", "authorname:lippmanrasmussen" ]

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    Section 5.4 Exercises

    1. If \(\theta =\frac{\pi \; }{4}\) , find exact values for \(\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\;\).

    2. If \(\theta =\frac{7\pi \; }{4}\) , find exact values for \(\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\;\).

    3. If \(\theta =\frac{5\pi \; }{6}\) , find exact values for \(\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\;\).

    4. If \(\theta =\frac{\pi \; }{6}\) , find exact values for \(\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\).

    5. If \(\theta =\frac{2\pi \; }{3}\) , find exact values for \(\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\;\).

    6. If \(\theta =\frac{4\pi \; }{3}\) , find exact values for \(\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\).

    7. Evaluate: a. \(\sec \left(135{}^\circ \right)\) b. \(\csc \left(210{}^\circ \right)\) c. \(\tan \left(60{}^\circ \right)\) d. \(\cot \left(225{}^\circ \right)\)

    8. Evaluate: a. \(\sec \left(30{}^\circ \right)\) b. \(\csc \left(315{}^\circ \right)\) c. \(\tan \left(135{}^\circ \right)\) d. \(\cot \left(150{}^\circ \right)\)

    9. If \(\sin \left(\theta \right)=\frac{3}{4}\), and \(\theta\) is in quadrant II, find \(\cos \left(\theta \right),\; \sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\).

    10. If \(\sin \left(\theta \right)=\frac{2}{7}\), and \(\theta\) is in quadrant II, find \(\cos \left(\theta \right),\; \sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\).

    11. If \(\cos \left(\theta \right)=-\frac{1}{3}\), and \(\theta\) is in quadrant III, find \(\sin \left(\theta \right),\; \sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\).

    12. If \(\cos \left(\theta \right)=\frac{1}{5}\), and \(\theta\) is in quadrant I, find \(\sin \left(\theta \right),\; \sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\).

    13. If \(\tan \left(\theta \right)=\frac{12}{5}\), and \(0\le \theta <\frac{\pi }{2}\), find \(\sin \left(\theta \right),\; \cos \left(\theta \right),\sec \left(\theta \right),\; \csc \left(\theta \right),\; \cot \left(\theta \right)\).

    14. If \(\tan \left(\theta \right)=4\), and \(0\le \theta <\frac{\pi }{2}\), find \(\sin \left(\theta \right),\; \cos \left(\theta \right),\sec \left(\theta \right),\; \csc \left(\theta \right),\; \cot \left(\theta \right)\).

    1. Use a calculator to find sine, cosine, and tangent of the following values:

    a. 0.15 b. 4 c. 70\(\mathrm{{}^\circ}\) d. 283\(\mathrm{{}^\circ}\)

    1. Use a calculator to find sine, cosine, and tangent of the following values:

    a. 0.5 b. 5.2 c. 10\(\mathrm{{}^\circ}\) d. 195\(\mathrm{{}^\circ}\)

    Simplify each of the following to an expression involving a single trig function with no fractions.

    1. \(\csc (t)\tan \left(t\right)\)

    2. \(\cos (t)\csc \left(t\right)\)

    3. \(\frac{\sec \left(t\right)}{\csc \left(t\right)\; }\)

    4. \(\frac{\cot \left(t\right)}{\csc \left(t\right)}\)

    5. \(\frac{\sec \left(t\right)-\cos \left(t\right)}{\sin \left(t\right)}\)

    6. \(\frac{\tan \left(t\right)}{\sec \left(t\right)-\cos \left(t\right)}\)

    7. \(\frac{1+\cot \left(t\right)}{1+\tan \left(t\right)}\)

    8. \(\frac{1+\sin \left(t\right)}{1+\csc \left(t\right)}\)

    9. \(\frac{\sin ^{2} \left(t\right)+\cos ^{2} \left(t\right)}{\cos ^{2} \left(t\right)}\)

    10. \(\frac{1-\sin ^{2} \left(t\right)}{\sin ^{2} \left(t\right)}\)

    Prove the identities.

    1. \(\frac{\sin ^{2} \left(\theta \right)}{1+\cos \left(\theta \right)} =1-\cos \left(\theta \right)\)

    2. image

    3. image

    4. image

    5. image

    6. image

    7. image

    8. image

    9. image

    10. image

    11. image

    12. image

    Section 5.5 Right Triangle Trigonometry 391