Math / Trigonometry

Questionolve for the indicat

Studdy Solution

STEP 1

1. The triangle is a right triangle.
2. The angle given is $32^\circ$ .
3. The hypotenuse of the triangle is $4$ meters.
4. We need to find the length of the side opposite the $32^\circ$ angle, labeled as $x$ .

STEP 2

1. Identify the trigonometric function to use.
2. Set up the equation using the trigonometric function.
3. Solve for the unknown side $x$ .

STEP 3

Identify the trigonometric function to use:
Since we need to find the side opposite the given angle and we know the hypotenuse, we use the sine function. Recall that:
$\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$

STEP 4

Set up the equation using the sine function:
$\sin(32^\circ) = \frac{x}{4}$

STEP 5

Solve for the unknown side $x$ :
Multiply both sides by $4$ to isolate $x$ :
$x = 4 \times \sin(32^\circ)$
Calculate the value of $x$ using a calculator:
$x \approx 4 \times 0.5299$
$x \approx 2.1196$
Thus, the length of the side opposite the $32^\circ$ angle is approximately:
$\boxed{2.12 \text{ meters}}$

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