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find the exact values of the five remaining trigonometric functions of …

Question

find the exact values of the five remaining trigonometric functions of \\(\theta\\).

  1. \\(\tan \theta = 2\\), where \\(\sin \theta > 0\\) and \\(\cos \theta > 0\\)
  2. \\(\csc \theta = 2\\), where \\(\sin \theta > 0\\) and \\(\cos \theta < 0\\)
  3. \\(\sin \theta = -\frac{1}{5}\\), where \\(\cos \theta > 0\\)
  4. \\(\cos \theta = -\frac{12}{13}\\), where \\(\sin \theta < 0\\)
  5. \\(\sec \theta = \sqrt{3}\\), where \\(\sin \theta < 0\\) and \\(\cos \theta > 0\\)
  6. \\(\cot \theta = 1\\), where \\(\sin \theta < 0\\) and \\(\cos \theta < 0\\)
  7. \\(\tan \theta = -1\\), where \\(\sin \theta < 0\\)
  8. \\(\cos \theta = -\frac{1}{2}\\), where \\(\sin \theta > 0\\)

Explanation:

Identify the quadrant for each question

Using the Trigonometric Functions of General Angles knowledge point:

  • Q33: \(\tan \theta = 2\), \(\sin \theta > 0\), \(\cos \theta > 0\) \(\implies\) Quadrant I.
  • Q34: \(\csc \theta = 2\), \(\sin \theta > 0\), \(\cos \theta < 0\) \(\implies\) Quadrant II.
  • Q35: \(\sin \theta = -\frac{1}{5}\), \(\cos \theta > 0\) \(\implies\) Quadrant IV.
  • Q36: \(\cos \theta = -\frac{12}{13}\), \(\sin \theta < 0\) \(\implies\) Quadrant III.
  • Q37: \(\sec \theta = \sqrt{3}\), \(\sin \theta < 0\), \(\cos \theta > 0\) \(\implies\) Quadrant IV.
  • Q38: \(\cot \theta = 1\), \(\sin \theta < 0\), \(\cos \theta < 0\) \(\implies\) Quadrant III.
  • Q39: \(\tan \theta = -1\), \(\sin \theta < 0\) \(\implies\) Quadrant IV.
  • Q40: \(\cos \theta = -\frac{1}{2}\), \(\sin \theta > 0\) \(\implies\) Quadrant II.

Solve Questions 33 to 34

Using the Trigonometric Ratios and Exact Trigonometric Values knowledge points:

  • Q33: Quadrant I. Let \(y = 2\), \(x = 1\). Then \(r = \sqrt{1^2 + 2^2} = \sqrt{5}\).
$$ \sin \theta = \frac{2\sqrt{5}}{5}, \cos \theta = \frac{\sqrt{5}}{5}, \csc \theta = \frac{\sqrt{5}}{2}, \sec \theta = \sqrt{5}, \cot \theta = \frac{1}{2} $$
  • Q34: Quadrant II. \(\csc \theta = 2 \implies \sin \theta = \frac{1}{2}\). Let \(y = 1\), \(r = 2\). Then \(x = -\sqrt{2^2 - 1^2} = -\sqrt{3}\).
$$ \sin \theta = \frac{1}{2}, \cos \theta = -\frac{\sqrt{3}}{2}, \tan \theta = -\frac{\sqrt{3}}{3}, \sec \theta = -\frac{2\sqrt{3}}{3}, \cot \theta = -\sqrt{3} $$

Solve Questions 35 to 36

Using the Trigonometric Ratios and Exact Trigonometric Values knowledge points:

  • Q35: Quadrant IV. \(\sin \theta = -\frac{1}{5}\). Let \(y = -1\), \(r = 5\). Then \(x = \sqrt{5^2 - (-1)^2} = \sqrt{24} = 2\sqrt{6}\).
$$ \cos \theta = \frac{2\sqrt{6}}{5}, \tan \theta = -\frac{\sqrt{6}}{12}, \csc \theta = -5, \sec \theta = \frac{5\sqrt{6}}{12}, \cot \theta = -2\sqrt{6} $$
  • Q36: Quadrant III. \(\cos \theta = -\frac{12}{13}\). Let \(x = -12\), \(r = 13\). Then \(y = -\sqrt{13^2 - (-12)^2} = -5\).
$$ \sin \theta = -\frac{5}{13}, \tan \theta = \frac{5}{12}, \csc \theta = -\frac{13}{5}, \sec \theta = -\frac{13}{12}, \cot \theta = \frac{12}{5} $$

Solve Questions 37 to 38

Using the Trigonometric Ratios and Exact Trigonometric Values knowledge points:

  • Q37: Quadrant IV. \(\sec \theta = \sqrt{3} \implies \cos \theta = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\). Let \(x = 1\), \(r = \sqrt{3}\). Then \(y = -\sqrt{(\sqrt{3})^2 - 1^2} = -\sqrt{2}\).
$$ \sin \theta = -\frac{\sqrt{6}}{3}, \cos \theta = \frac{\sqrt{3}}{3}, \tan \theta = -\sqrt{2}, \csc \theta = -\frac{\sqrt{6}}{2}, \cot \theta = -\frac{\sqrt{2}}{2} $$
  • Q38: Quadrant III. \(\cot \theta = 1\). Let \(x = -1\), \(y = -1\). Then \(r = \sqrt{(-1)^2 + (-1)^2} = \sqrt{2}\).

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\sin \theta = -\frac{\sqrt{2}}{2}, \cos \theta = -\frac{\…

Answer:

Question 33

$$ \sin \theta = \frac{2\sqrt{5}}{5}, \quad \cos \theta = \frac{\sqrt{5}}{5}, \quad \csc \theta = \frac{\sqrt{5}}{2}, \quad \sec \theta = \sqrt{5}, \quad \cot \theta = \frac{1}{2} $$

Question 34

$$ \sin \theta = \frac{1}{2}, \quad \cos \theta = -\frac{\sqrt{3}}{2}, \quad \tan \theta = -\frac{\sqrt{3}}{3}, \quad \sec \theta = -\frac{2\sqrt{3}}{3}, \quad \cot \theta = -\sqrt{3} $$

Question 35

$$ \cos \theta = \frac{2\sqrt{6}}{5}, \quad \tan \theta = -\frac{\sqrt{6}}{12}, \quad \csc \theta = -5, \quad \sec \theta = \frac{5\sqrt{6}}{12}, \quad \cot \theta = -2\sqrt{6} $$

Question 36

$$ \sin \theta = -\frac{5}{13}, \quad \tan \theta = \frac{5}{12}, \quad \csc \theta = -\frac{13}{5}, \quad \sec \theta = -\frac{13}{12}, \quad \cot \theta = \frac{12}{5} $$

Question 37

$$ \sin \theta = -\frac{\sqrt{6}}{3}, \quad \cos \theta = \frac{\sqrt{3}}{3}, \quad \tan \theta = -\sqrt{2}, \quad \csc \theta = -\frac{\sqrt{6}}{2}, \quad \cot \theta = -\frac{\sqrt{2}}{2} $$

Question 38

$$ \sin \theta = -\frac{\sqrt{2}}{2}, \quad \cos \theta = -\frac{\sqrt{2}}{2}, \quad \tan \theta = 1, \quad \csc \theta = -\sqrt{2}, \quad \sec \theta = -\sqrt{2} $$

Question 39

$$ \sin \theta = -\frac{\sqrt{2}}{2}, \quad \cos \theta = \frac{\sqrt{2}}{2}, \quad \csc \theta = -\sqrt{2}, \quad \sec \theta = \sqrt{2}, \quad \cot \theta = -1 $$

Question 40

$$ \sin \theta = \frac{\sqrt{3}}{2}, \quad \tan \theta = -\sqrt{3}, \quad \csc \theta = \frac{2\sqrt{3}}{3}, \quad \sec \theta = -2, \quad \cot \theta = -\frac{\sqrt{3}}{3} $$