Question

there is a right triangle, the hypotenuse length is 11, one of the acute angles is 21 degrees, and the length of the right - angled side opposite to the 21 - degree angle is x, find the value of x.

Explanation:

Step1: Identify trigonometric function

We have a right triangle with hypotenuse \( 11 \) and angle \( 21^\circ \). The side \( x \) is opposite the \( 21^\circ \) angle. So we use the sine function: \( \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} \).

Step2: Substitute values

Substitute \( \theta = 21^\circ \), opposite \( = x \), hypotenuse \( = 11 \) into the sine formula: \( \sin(21^\circ)=\frac{x}{11} \).

Step3: Solve for \( x \)

Multiply both sides by \( 11 \): \( x = 11\times\sin(21^\circ) \). Calculate \( \sin(21^\circ)\approx0.3584 \), so \( x\approx11\times0.3584 = 3.9424 \).

Answer:

\( x\approx3.94 \) (or more precise \( 3.9424 \))