Reduction Formulas — Hard Practice Quiz

A Trigonometry cheat sheet for Reduction Formulas — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Formulas & key concepts

Negative Angle

$$\sin(-\alpha) = -\sin \alpha, \quad \cos(-\alpha) = \cos \alpha$$

Cofunction (90° - α)

$$\sin(90^\circ - \alpha) = \cos \alpha, \quad \cos(90^\circ - \alpha) = \sin \alpha$$

90° + α

$$\sin(90^\circ + \alpha) = \cos \alpha, \quad \cos(90^\circ + \alpha) = -\sin \alpha$$

Supplementary Angle (180° - α)

$$\sin(180^\circ - \alpha) = \sin \alpha, \quad \cos(180^\circ - \alpha) = -\cos \alpha$$

180° + α

$$\sin(180^\circ + \alpha) = -\sin \alpha, \quad \cos(180^\circ + \alpha) = -\cos \alpha$$

Practice quiz

  1. Simplify the expression $ \sin(270^\circ - \alpha) $.

    • $ \sin \alpha $
    • $ \cos \alpha $
    • $ -\sin \alpha $
    • $ -\cos \alpha $

    Answer: $ -\cos \alpha $

  2. Simplify the expression $ \cos(270^\circ + \alpha) $.

    • $ \sin \alpha $
    • $ -\sin \alpha $
    • $ \cos \alpha $
    • $ -\cos \alpha $

    Answer: $ \sin \alpha $

  3. If $ \sin \alpha = 0.6 $ and $ \alpha $ is in the first quadrant, what is the value of $ \cos(180^\circ - \alpha) + \sin(90^\circ + \alpha) $?

    • $ 0 $
    • $ 1.6 $
    • $ -1.6 $
    • $ 0.8 $

    Answer: $ 0 $

  4. Given $ \sin(90^\circ - \theta) = 2\cos(180^\circ + \theta) $, find the value of $ \tan \theta $.

    • $ 0 $
    • $ 1 $
    • $ -1 $
    • Undefined

    Answer: Undefined

  5. Which of the following expressions is NOT equivalent to $ \sin \alpha $?

    • $ \cos(90^\circ - \alpha) $
    • $ -\sin(-\alpha) $
    • $ \sin(180^\circ - \alpha) $
    • $ \cos(90^\circ + \alpha) $

    Answer: $ \cos(90^\circ + \alpha) $

  6. Simplify the expression $ \tan(180^\circ + (90^\circ - \alpha)) $.

    • $ \tan \alpha $
    • $ -\tan \alpha $
    • $ \cot \alpha $
    • $ -\cot \alpha $

    Answer: $ \cot \alpha $

  7. If $ \cos \theta = k $, where $ \theta $ is an acute angle, express $ \sin(180^\circ + \theta) + \cos(90^\circ + \theta) $ in terms of $ k $.

    • $ 0 $
    • $ -2k $
    • $ -2\sqrt{1 - k^2} $
    • $ 2\sqrt{1 - k^2} $

    Answer: $ -2\sqrt{1 - k^2} $

  8. Solve for $ \alpha $ in the range $ 0^\circ \le \alpha < 360^\circ $ if $ \sin(90^\circ + \alpha) = \cos(180^\circ - \alpha) $.

    • $ 0^\circ, 180^\circ $
    • $ 90^\circ, 270^\circ $
    • $ 45^\circ, 225^\circ $
    • $ 135^\circ, 315^\circ $

    Answer: $ 90^\circ, 270^\circ $

  9. Simplify the expression $ \frac{\sin(180^\circ - \alpha) \cdot \cos(90^\circ + \alpha)}{\cos(-\alpha) \cdot \sin(180^\circ + \alpha)} $.

    • $ \tan \alpha $
    • $ \cot \alpha $
    • $ -\tan \alpha $
    • $ -\cot \alpha $

    Answer: $ \tan \alpha $

  10. If $ \cos \theta = -0.8 $ and $ \theta $ is in the third quadrant, what is the value of $ \sin(90^\circ + \theta) - \cos(180^\circ - \theta) $?

    • $ 0 $
    • $ -1.6 $
    • $ 1.6 $
    • $ -0.8 $

    Answer: $ -1.6 $

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