Definitions — Practice Quiz

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

Formulas & key concepts

Sine and Cosine

$$\sin \alpha = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos \alpha = \frac{\text{adjacent}}{\text{hypotenuse}}$$

Tangent and Cotangent

$$\tan \alpha = \frac{\text{opposite}}{\text{adjacent}}, \quad \cot \alpha = \frac{\text{adjacent}}{\text{opposite}}$$

Secant and Cosecant

$$\sec \alpha = \frac{\text{hypotenuse}}{\text{adjacent}}, \quad \csc \alpha = \frac{\text{hypotenuse}}{\text{opposite}}$$

Practice quiz

  1. In a right-angled triangle, if the opposite side to angle $\theta$ is $3$ and the hypotenuse is $5$, what is $\sin \theta$?

    • $3/5$
    • $4/5$
    • $5/3$
    • $3/4$

    Answer: $3/5$

  2. A right triangle has an adjacent side of length $8$ and a hypotenuse of length $17$ with respect to angle $\beta$. What is $\cos \beta$?

    • $8/17$
    • $15/17$
    • $17/8$
    • $8/15$

    Answer: $8/17$

  3. For an angle $\phi$ in a right triangle, the opposite side is $12$ and the adjacent side is $5$. What is $\tan \phi$?

    • $5/12$
    • $12/5$
    • $13/5$
    • $5/13$

    Answer: $12/5$

  4. If the adjacent side to angle $\gamma$ is $7$ and the opposite side is $24$, what is $\cot \gamma$?

    • $24/7$
    • $7/24$
    • $7/25$
    • $24/25$

    Answer: $7/24$

  5. In a right triangle, the hypotenuse is $10$ and the adjacent side to angle $\alpha$ is $6$. What is $\sec \alpha$?

    • $6/10$
    • $10/6$
    • $8/10$
    • $10/8$

    Answer: $10/6$

  6. Given a right triangle where the hypotenuse is $25$ and the opposite side to angle $\delta$ is $7$. What is $\csc \delta$?

    • $7/25$
    • $25/7$
    • $24/25$
    • $25/24$

    Answer: $25/7$

  7. If $\sin \theta = \frac{1}{2}$, what is $\csc \theta$?

    • $2$
    • $\frac{1}{2}$
    • $\sqrt{3}/2$
    • $1/\sqrt{3}$

    Answer: $2$

  8. If $\cos \alpha = \frac{3}{5}$, what is $\sec \alpha$?

    • $3/5$
    • $5/3$
    • $4/5$
    • $5/4$

    Answer: $5/3$

  9. If $\tan \beta = \frac{4}{3}$, what is $\cot \beta$?

    • $4/3$
    • $3/4$
    • $5/3$
    • $3/5$

    Answer: $3/4$

  10. In a right triangle, the opposite side to angle $\theta$ is $5$ and the hypotenuse is $13$. What is $\tan \theta$?

    • $5/13$
    • $12/13$
    • $5/12$
    • $12/5$

    Answer: $5/12$

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