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Mathematics LibreTexts

5.4.4E: The Other Trigonometric Functions (Exercises)

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    Section 5.4 Exercise

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

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

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

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

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

    6. If \(\theta =\dfrac{4\pi \; }{3}\) , 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 \(\sin \left(\theta \right)=\dfrac{3}{4}\), 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 \(\sin \left(\theta \right)=\dfrac{2}{7}\), 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 \(\cos \left(\theta \right)=-\dfrac{1}{3}\), 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 \(\cos \left(\theta \right)=\dfrac{1}{5}\), 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 \(\tan \left(\theta \right)=\dfrac{12}{5}\), and \(0\le \theta <\dfrac{\pi }{2}\), 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 \(0\le \theta <\dfrac{\pi }{2}\), 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. \(\dfrac{\sec \left(t\right)}{\csc \left(t\right)\; }\)

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

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

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

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

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

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

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

    Prove the identities.

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

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

    Answer

    1. \(\text{sec} (\theta) = \sqrt{2}\), \(\text{csc} (\theta) = \sqrt{2}\), \(\text{tan} (\theta) = 1\), \(\text{cot} (\theta) = 1\)

    3. \(\text{sec} (\theta) = -\dfrac{2\sqrt{3}}{3}\), \(\text{csc} (\theta) = 2\), \(\text{tan} (\theta) = -\dfrac{\sqrt{3}}{3}\), \(\text{cot} (\theta) = -\sqrt{3}\)

    5. \(\text{sec} (\theta) = -2\), \(\text{csc} (\theta) = \dfrac{2\sqrt{3}}{3}\), \(\text{tan} (\theta) = -\sqrt{3}\), \(\text{cot} (\theta) = -\dfrac{\sqrt{3}}{3}\)

    7. a. \(\text{sec} (135^{\circ}) = -\sqrt{2}\)
    b. \(\text{csc} (210^{\circ}) = -2\)
    c. \(\text{tan} (60^{\circ}) = \sqrt{3}\)
    d. \(\text{cot} (225^{\circ}) = 1\)

    9. \(\cos(\theta) = -\dfrac{\sqrt{7}}{4}\), \(\sec (\theta) = -\dfrac{4\sqrt{7}}{7}\), \(\csc(\theta) = \dfrac{4}{3}\), \(\tan(\theta) = -\dfrac{3\sqrt{7}}{7}\), \(\cot(\theta) = -\dfrac{\sqrt{7}}{3}\)

    11. \(\sin(\theta) = -\dfrac{2\sqrt{2}}{3}\), \(\csc(\theta) = -\dfrac{3\sqrt{2}}{3}\), \(\sec(\theta) = -3\), \(\tan(\theta) = 2\sqrt{2}\), \(\cot(\theta) = \dfrac{\sqrt{2}}{4}\)

    13. \(\sin(\theta) = \dfrac{12}{13}\), \(\cos(\theta) = \dfrac{5}{13}\), \(\sec(\theta) = \dfrac{13}{5}\), \(\csc(\theta) = \dfrac{13}{12}\), \(\cot(\theta) = \dfrac{5}{12}\)

    15. a. sin(0.15) = 0.1494 cos(0.15) = 0.9888 tan(0.15) = 0.1511
    b. sin(4) = -0.7568 cos(4) = -0.6536 tan(4) = 1.1578
    c. sin(\(70^{\circ}\)) = 0.9397 cos(\(70^{\circ}\)) = 0.3420 tan(\(70^{\circ}\)) = 2.7475
    d. sin(\(283^{\circ}\)) = -0.9744 cos(\(283^{\circ}\)) = 0.2250 tan(\(283^{\circ}\)) = -4.3315

    17. sec(\(t\))

    19. tan(\(t\))

    21. tan(\(t\))

    23. cot(\(t\))

    25. \((\sec(t))^2\)