Definitions — Hard Practice Quiz

A Trigonometry cheat sheet for Definitions — every key formula with its symbols defined — plus a hard-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. If $\sin \theta = x$ and $\cos \theta = y$ for an acute angle $\theta$, what is $\tan \theta$?

    • $\frac{x}{y}$
    • $\frac{y}{x}$
    • $\frac{1}{x}$
    • $\frac{1}{y}$

    Answer: $\frac{x}{y}$

  2. Given $\sec \alpha = A$ and $\csc \alpha = B$ for an acute angle $\alpha$, express $\tan \alpha$ in terms of $A$ and $B$.

    • $\frac{A}{B}$
    • $\frac{B}{A}$
    • $\frac{1}{AB}$
    • $\sqrt{A^2 - B^2}$

    Answer: $\frac{A}{B}$

  3. In a right-angled triangle, the hypotenuse measures $10$ units. If $\tan \theta = \frac{3}{4}$, what is the length of the side opposite to $\theta$?

    • $6$ units
    • $8$ units
    • $7.5$ units
    • $5$ units

    Answer: $6$ units

  4. If the length of the side opposite to an acute angle $\alpha$ in a right triangle is doubled, while the length of the adjacent side remains constant, how does $\cot \alpha$ change?

    • It is doubled.
    • It is halved.
    • It remains constant.
    • It is quadrupled.

    Answer: It is halved.

  5. For an acute angle $\theta$, express $\csc \theta$ in terms of $\tan \theta$.

    • $\frac{\sqrt{\tan^2 \theta + 1}}{\tan \theta}$
    • $\frac{\tan \theta}{\sqrt{\tan^2 \theta + 1}}$
    • $\sqrt{\tan^2 \theta + 1}$
    • $\frac{1}{\tan \theta}$

    Answer: $\frac{\sqrt{\tan^2 \theta + 1}}{\tan \theta}$

  6. A ladder leans against a vertical wall, making an angle $\alpha$ with the horizontal ground. The base of the ladder is $5$ meters from the wall. If $\sec \alpha = 2$, what is the height the ladder reaches on the wall?

    • $5\sqrt{3}$ meters
    • $10$ meters
    • $5$ meters
    • $2.5\sqrt{3}$ meters

    Answer: $5\sqrt{3}$ meters

  7. Given $\sin \theta = \frac{1}{3}$ for an acute angle $\theta$, find the value of the expression $\frac{\tan \theta + \cot \theta}{\sec \theta \cdot \csc \theta}$.

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

    Answer: $1$

  8. As an acute angle $\alpha$ increases from $0$ to $\frac{\pi}{2}$ (but not including the endpoints), how does the product $\tan \alpha \cdot \cot \alpha$ change?

    • It increases.
    • It decreases.
    • It remains constant.
    • It first increases then decreases.

    Answer: It remains constant.

  9. In a right triangle, the side adjacent to an acute angle $\beta$ has length $A$. If $\csc \beta = 2.5$, what is the length of the side opposite to $\beta$ in terms of $A$?

    • $\frac{2A}{\sqrt{21}}$
    • $\frac{A}{\sqrt{21}}$
    • $\frac{5A}{2}$
    • $\frac{2A}{5}$

    Answer: $\frac{2A}{\sqrt{21}}$

  10. Simplify the expression: $(\sin \theta + \cos \theta)^2 - 2 \tan \theta \cos^2 \theta$.

    • $1$
    • $\sin^2 \theta$
    • $\cos^2 \theta$
    • $\sin \theta \cos \theta$

    Answer: $1$

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