Question

trigonometry quiz
find the missing side. round to the nearest tenth.
1)
2)
find the measure of the indicated angle to the nearest degree.
3)

Explanation:

Problem 1: Find the missing side \( x \)

Step 1: Identify the trigonometric ratio

We have a right triangle with an angle of \( 49^\circ \), the opposite side to \( 49^\circ \) is \( 16 \), and the adjacent side is \( x \). So we use the tangent function: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)
\( \tan(49^\circ) = \frac{16}{x} \)

Step 2: Solve for \( x \)

Rearrange the formula to solve for \( x \): \( x = \frac{16}{\tan(49^\circ)} \)
Calculate \( \tan(49^\circ) \approx 1.1504 \)
\( x = \frac{16}{1.1504} \approx 13.9 \)

Step 1: Identify the trigonometric ratio

We have a right triangle with an angle of \( 68^\circ \), the hypotenuse is \( 19 \), and the adjacent side to \( 68^\circ \) is \( x \). So we use the cosine function: \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)
\( \cos(68^\circ) = \frac{x}{19} \)

Step 2: Solve for \( x \)

Rearrange the formula to solve for \( x \): \( x = 19 \times \cos(68^\circ) \)
Calculate \( \cos(68^\circ) \approx 0.3746 \)
\( x = 19 \times 0.3746 \approx 7.1 \)

Step 1: Identify the trigonometric ratio

We have a right triangle with the adjacent side \( 26 \) and the hypotenuse \( 40 \). So we use the cosine function: \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)
\( \cos(\theta) = \frac{26}{40} = 0.65 \)

Step 2: Solve for \( \theta \)

Take the inverse cosine (arccos) of \( 0.65 \): \( \theta = \arccos(0.65) \)
Calculate \( \arccos(0.65) \approx 49^\circ \) (rounded to the nearest degree)

Answer:

\( x \approx 13.9 \)

Problem 2: Find the missing side \( x \)