Reduction Formulas — Practice Quiz

A Trigonometry cheat sheet for Reduction Formulas — every key formula with its symbols defined — plus a medium-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 $\sin(-30^\circ)$.

    • $\sin(30^\circ)$
    • $-\sin(30^\circ)$
    • $\cos(30^\circ)$
    • $-\cos(30^\circ)$

    Answer: $-\sin(30^\circ)$

  2. Which of the following is equivalent to $\cos(-60^\circ)$?

    • $-\cos(60^\circ)$
    • $\cos(60^\circ)$
    • $\sin(60^\circ)$
    • $-\sin(60^\circ)$

    Answer: $\cos(60^\circ)$

  3. If $\sin(20^\circ) = p$, what is the value of $\cos(70^\circ)$?

    • $p$
    • $-p$
    • $\frac{1}{p}$
    • $\sqrt{1-p^2}$

    Answer: $p$

  4. Simplify the expression $\sin(90^\circ - 45^\circ)$.

    • $\sin(45^\circ)$
    • $-\sin(45^\circ)$
    • $\cos(45^\circ)$
    • $-\cos(45^\circ)$

    Answer: $\cos(45^\circ)$

  5. Express $\sin(120^\circ)$ using the identity for $\sin(90^\circ + \alpha)$.

    • $\sin(30^\circ)$
    • $-\sin(30^\circ)$
    • $\cos(30^\circ)$
    • $-\cos(30^\circ)$

    Answer: $\cos(30^\circ)$

  6. If $\cos(90^\circ + \theta) = -0.5$, what is the value of $\sin(\theta)$?

    • $0.5$
    • $-0.5$
    • $\sqrt{1-0.5^2}$
    • $-\sqrt{1-0.5^2}$

    Answer: $0.5$

  7. Simplify the expression $\sin(180^\circ - 60^\circ)$.

    • $\sin(60^\circ)$
    • $-\sin(60^\circ)$
    • $\cos(60^\circ)$
    • $-\cos(60^\circ)$

    Answer: $\sin(60^\circ)$

  8. Which expression is equivalent to $\cos(180^\circ - \beta)$?

    • $\cos(\beta)$
    • $-\cos(\beta)$
    • $\sin(\beta)$
    • $-\sin(\beta)$

    Answer: $-\cos(\beta)$

  9. Given $\sin(180^\circ + x) = 0.8$, what is the value of $\sin(x)$?

    • $0.8$
    • $-0.8$
    • $\sqrt{1-0.8^2}$
    • $-\sqrt{1-0.8^2}$

    Answer: $-0.8$

  10. Simplify $\cos(210^\circ)$ using the identity for $\cos(180^\circ + \alpha)$.

    • $\cos(30^\circ)$
    • $-\cos(30^\circ)$
    • $\sin(30^\circ)$
    • $-\sin(30^\circ)$

    Answer: $-\cos(30^\circ)$

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