Math  /  Trigonometry

Question22. Find the measure of θ\theta in the given figure.

Studdy Solution

STEP 1

1. The triangle is a right triangle.
2. The side opposite the right angle is the hypotenuse.
3. The side adjacent to θ\theta is 3.
4. The hypotenuse is 7.62.

STEP 2

1. Recall the trigonometric ratio for cosine.
2. Set up the equation using the cosine ratio.
3. Solve for θ\theta.

STEP 3

Recall the trigonometric ratio for cosine in a right triangle:
cos(θ)=Adjacent sideHypotenuse \cos(\theta) = \frac{\text{Adjacent side}}{\text{Hypotenuse}}

STEP 4

Set up the equation using the cosine ratio:
cos(θ)=37.62 \cos(\theta) = \frac{3}{7.62}

STEP 5

Solve for θ\theta by taking the inverse cosine:
θ=cos1(37.62) \theta = \cos^{-1}\left(\frac{3}{7.62}\right)
Calculate θ\theta using a calculator:
θcos1(0.3932) \theta \approx \cos^{-1}(0.3932)
θ67.02 \theta \approx 67.02^\circ
The measure of θ\theta is approximately:
67.02 \boxed{67.02^\circ}

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