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Mathematics LibreTexts

5.4E: The Other Trigonometric Functions (Exercises)

  • Page ID
    13915
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    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 \(\sin \left(\theta \right),\; \cos \left(\theta \right),\sec \left(\theta \right),\; \csc \left(\theta \right),\; \cot \left(\theta \right)\).

    15. Use a calculator to find sine, cosine, and tangent of the following values:
      a. 0.15
      b. 4
      c. 70\(\mathrm{{}^\circ}\)
      d. 283\(\mathrm{{}^\circ}\)

    16. 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.

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

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

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

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

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

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

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

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

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

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

    Prove the identities.

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

    28. \(\text{tan}^{2} (t) = \dfrac{1}{\text{cos}^(t)} - 1\)

    29. \(\text{sec}(a) - \text{cos}(a) = \text{sin}(a) \text{tan}(a)\)

    30. \(\dfrac{1 + \text{tan}^2(b)}{\text{tan}^2(b)} = \text{csc}^2(b)\)

    31. \(\dfrac{\text{csc}^2 (x) - \text{sin}^2 (x)}{\text{csc} (x) + \text{sin} (x)} = \text{cos} (x) \text{cot} (x)\)

    32. \(\dfrac{\text{sin} (\theta) - \text{cos} (\theta)}{\text{sec}(\theta) - \text{csc} (\theta)} = \text{sin} (\theta) \text{cos} (\theta)\)

    33. \(\dfrac{\text{csc}^2 (\alpha) - 1}{\text{csc}^2 (\alpha) - \text{csc} (\alpha)} = 1 + \text{sin} (\alpha)\)

    34. \(1 + \text{cot} (x) = \text{cos} (x) (\text{sec}(x) + \text{csc} (x))\)

    35. \(\dfrac{1 + \text{cos} (u)}{\text{sin} (u)} = \dfrac{\text{sin} (u)}{1 - \text{cos}(u)}\)

    36. \(2 \text{sec}^2 (t) = \dfrac{1 - \text{sin}(t)}{\text{cos}^2 (t)} + \dfrac{1}{1 - s \text{sin} (t)}\)

    37. \(\dfrac{\text{sin}^4 (\gamma) - \text{cos}^4 (\gamma)}{\text{sin} (\gamma) - \text{cos} (\gamma)} = \text{sin} (\gamma) + \text{cos} (\gamma)\)

    38. \(\dfrac{(1 + \text{cos}(A))(1 - \text{cos} (A))}{\text{sin} (A)} = \text{sin} (A)\)