Trigonometry Questions Using Unit Circle Symmetry

This page contains multiple-choice questions that test your understanding of unit circle symmetry and its use in evaluating trigonometric functions. Complete solutions and explanations are provided in the answers section.

Review of Symmetries of the Unit Circle

1. Symmetry about the x-axis \[ \sin(-\theta) = -\sin(\theta), \quad \cos(-\theta) = \cos(\theta) \] 2. Symmetry about the y-axis \[ \cos(\pi - \theta) = -\cos(\theta), \quad \sin(\pi - \theta) = \sin(\theta) \] 3. Symmetry about the origin \[ \cos(\theta \pm \pi) = -\cos(\theta), \quad \sin(\theta \pm \pi) = -\sin(\theta) \]

Questions

Question 1

If \(0 < t < \frac{\pi}{2}\) and \(\sin t = 0.35\), find:

\[ \cos(t + \pi) = ? \]

a) 0.94
b) −0.94
c) 0.81
d) −0.81

Question 2

If \(\tan t = 13\), then:

\[ \cot(-t) = ? \]

a) 13
b) \( \frac{1}{13} \)
c) \( -\frac{1}{13} \)
d) −13

Question 3

Evaluate:

\[ \cos x + \cos(\pi - x) = ? \]

a) \(2\cos x\)
b) \(\cos x - \sin x\)
c) \(\cos x + \sin x\)
d) 0

Question 4

If \(0 < t < \frac{\pi}{2}\) and \(\sin t = 0.65\), find:

\[ \sin(t + \pi) = ? \]

a) −0.65
b) 0.65
c) 0.35
d) 0.76

Question 5

If \(\cos(-t) = 0.34\), then:

\[ \cos t = ? \]

a) −0.34
b) 0.66
c) 0.34
d) −0.66

Question 6

If \(\sin(-t) = 0.54\), find:

\[ - \sin t = ? \]

a) 0.54
b) −0.54
c) −0.46
d) 0.46

Question 7

Which of the following is not equal to \(\tan t\)?

a) −\(\tan(-t)\)
b) \(\tan(t + 2\pi)\)
c) \(\tan(t + \pi)\)
d) \(\tan\left(t + \frac{\pi}{2}\right)\)

Question 8

Which of the following identities is incorrect?

a) \(\sin x = -\sin(-x)\)
b) \(\sec(-t) = \sec t\)
c) \(\sin(\pi + x) = \sin x\)
d) \(\cos(\pi - x) = -\cos x\)

Answers and Explanations

  1. Answer: b)
    Since \(\cos(t + \pi) = -\cos t\) and \[ \cos t = \sqrt{1 - \sin^2 t} = \sqrt{1 - 0.35^2} \approx 0.94, \] we get: \[ \cos(t + \pi) = -0.94. \]
  2. Answer: c)
    \[ \cot(-t) = -\cot t = -\frac{1}{\tan t} = -\frac{1}{13}. \]
  3. Answer: d)
    Using the identity: \[ \cos(\pi - x) = -\cos x, \] we get: \[ \cos x + \cos(\pi - x) = \cos x - \cos x = 0. \]
  4. Answer: a)
    \[ \sin(t + \pi) = -\sin t = -0.65. \]
  5. Answer: c)
    Cosine is an even function: \[ \cos(-t) = \cos t, \] so \(\cos t = 0.34.\)
  6. Answer: a)
    Sine is odd: \[ \sin(-t) = -\sin t \Rightarrow -\sin t = 0.54. \]
  7. Answer: d)
    \(\tan(t + \pi/2)\) is undefined or changes value, so it is not equal to \(\tan t.\)
  8. Answer: c)
    \[ \sin(\pi + x) = -\sin x, \] so the given identity is incorrect.

Links and References

More Trigonometry Problems and Tutorials