Math / Geometry

Question41.

Studdy Solution

STEP 1

1. We are given a right triangle.
2. One angle is $52^\circ$ .
3. One side of the triangle is labeled as $35$ .
4. We need to find the length of the other sides.

STEP 2

1. Identify the given side relative to the $52^\circ$ angle.
2. Use trigonometric ratios to find the unknown sides.
3. Calculate the lengths of the unknown sides.

STEP 3

Determine which side is labeled as $35$ . Since the triangle is a right triangle, the side opposite the right angle is the hypotenuse. The side labeled $35$ could be either the adjacent side or the opposite side to the $52^\circ$ angle. We need this information to proceed.

STEP 4

Assume the side labeled $35$ is the adjacent side to the $52^\circ$ angle. This assumption will guide us in using the correct trigonometric ratio.

STEP 5

Use the cosine function to find the hypotenuse. The cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse:
$\cos(52^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}}$ $\cos(52^\circ) = \frac{35}{\text{hypotenuse}}$

STEP 6

Solve for the hypotenuse:
$\text{hypotenuse} = \frac{35}{\cos(52^\circ)}$

STEP 7

Use the sine function to find the opposite side. The sine of an angle in a right triangle is the ratio of the opposite side to the hypotenuse:
$\sin(52^\circ) = \frac{\text{opposite}}{\text{hypotenuse}}$

STEP 8

Solve for the opposite side using the hypotenuse found in Step 4:
$\text{opposite} = \sin(52^\circ) \times \text{hypotenuse}$ $\text{opposite} = \sin(52^\circ) \times \frac{35}{\cos(52^\circ)}$
The lengths of the unknown sides can be calculated using these formulas. You can use a calculator to find the numerical values.

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