Math / Trigonometry

QuestionWhich expressions can be used to find the measure of $\angle \theta$ ? Choose all that apply.

Studdy Solution

STEP 1

1. The triangle is a right triangle.
2. The angle $\theta$ is opposite side $z$ , adjacent to side $x$ , and the hypotenuse is $y$ .
3. We are looking for trigonometric expressions that relate $\theta$ to the sides of the triangle.

STEP 2

1. Identify the trigonometric ratios related to $\theta$ .
2. Write expressions using these ratios to find $\theta$ .
3. Verify which expressions are applicable.

STEP 3

Identify the trigonometric ratios for a right triangle: - Sine ( $\sin$ ) is opposite over hypotenuse. - Cosine ( $\cos$ ) is adjacent over hypotenuse. - Tangent ( $\tan$ ) is opposite over adjacent.

STEP 4

Write expressions using these ratios: - $\sin \theta = \frac{z}{y}$ - $\cos \theta = \frac{x}{y}$ - $\tan \theta = \frac{z}{x}$

STEP 5

Verify which expressions can be used to find $\theta$ : - $\theta = \sin^{-1} \left(\frac{z}{y}\right)$ - $\theta = \cos^{-1} \left(\frac{x}{y}\right)$ - $\theta = \tan^{-1} \left(\frac{z}{x}\right)$
The expressions that can be used to find the measure of $\angle \theta$ are:
$\boxed{\sin^{-1} \left(\frac{z}{y}\right), \cos^{-1} \left(\frac{x}{y}\right), \tan^{-1} \left(\frac{z}{x}\right)}$

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