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, the side opposite to angle $\theta$ is $8$ units long, and the hypotenuse is $17$ units long. What is the value of $\sin \theta$?

    • $\frac{8}{17}$
    • $\frac{15}{17}$
    • $\frac{8}{15}$
    • $\frac{17}{8}$

    Answer: $\frac{8}{17}$

  2. A ladder $10$ meters long leans against a wall. The base of the ladder is $6$ meters away from the wall. What is the cosine of the angle the ladder makes with the ground?

    • $\frac{10}{6}$
    • $\frac{6}{10}$
    • $\frac{8}{10}$
    • $\frac{6}{8}$

    Answer: $\frac{6}{10}$

  3. For a right triangle, if the side opposite to angle $\alpha$ is $12$ and the adjacent side is $5$, what is $\tan \alpha$?

    • $\frac{5}{12}$
    • $\frac{12}{5}$
    • $\frac{12}{13}$
    • $\frac{5}{13}$

    Answer: $\frac{12}{5}$

  4. Given a right triangle where the adjacent side to angle $\beta$ is $24$ and the opposite side is $7$, what is $\cot \beta$?

    • $\frac{7}{24}$
    • $\frac{24}{7}$
    • $\frac{7}{25}$
    • $\frac{24}{25}$

    Answer: $\frac{24}{7}$

  5. In a right triangle, the hypotenuse is $25$ units and the side adjacent to angle $\phi$ is $7$ units. What is $\sec \phi$?

    • $\frac{7}{25}$
    • $\frac{24}{25}$
    • $\frac{25}{7}$
    • $\frac{25}{24}$

    Answer: $\frac{25}{7}$

  6. If the hypotenuse of a right triangle is $13$ and the side opposite to angle $\gamma$ is $5$, what is $\csc \gamma$?

    • $\frac{5}{13}$
    • $\frac{12}{13}$
    • $\frac{13}{5}$
    • $\frac{13}{12}$

    Answer: $\frac{13}{5}$

  7. A kite string is $50$ meters long and makes an angle $\theta$ with the ground. If the kite is $30$ meters high, what is $\sin \theta$?

    • $\frac{3}{5}$
    • $\frac{4}{5}$
    • $\frac{3}{4}$
    • $\frac{5}{3}$

    Answer: $\frac{3}{5}$

  8. In a right triangle, if $\cos A = \frac{9}{41}$, what is $\tan A$?

    • $\frac{9}{40}$
    • $\frac{40}{9}$
    • $\frac{41}{9}$
    • $\frac{40}{41}$

    Answer: $\frac{40}{9}$

  9. A flagpole casts a shadow $16$ feet long. If the angle of elevation from the end of the shadow to the top of the flagpole is $\alpha$, and the flagpole is $12$ feet tall, what is $\cot \alpha$?

    • $\frac{12}{16}$
    • $\frac{16}{12}$
    • $\frac{12}{20}$
    • $\frac{16}{20}$

    Answer: $\frac{16}{12}$

  10. If $\csc x = \frac{29}{20}$ in a right triangle, what is the value of $\sec x$?

    • $\frac{20}{21}$
    • $\frac{21}{20}$
    • $\frac{29}{21}$
    • $\frac{21}{29}$

    Answer: $\frac{29}{21}$

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