Question

find the measure of each side indicated. round to the nearest tenth.
9)
b
11 37° x
c a
10)
b
x
a 32° 13 c
11)
a
5 50.1°
c x b
12)
b 11 a
60°
x
c

Explanation:

Problem 9

Step1: Identify trigonometric ratio

In right triangle \( \triangle ABC \), \( \angle C = 90^\circ \), \( \angle B = 37^\circ \), adjacent side to \( 37^\circ \) is \( 11 \), hypotenuse is \( x \). So \( \cos(37^\circ)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{11}{x} \).

Step2: Solve for \( x \)

Rearrange the formula: \( x = \frac{11}{\cos(37^\circ)} \). Calculate \( \cos(37^\circ)\approx0.8 \) (approximate value), so \( x=\frac{11}{0.8}=13.75\approx13.8 \) (using calculator for more accurate \( \cos(37^\circ)\approx0.7986 \), \( x=\frac{11}{0.7986}\approx13.8 \)).

Step1: Identify trigonometric ratio

In right triangle \( \triangle ABC \), \( \angle C = 90^\circ \), \( \angle A = 32^\circ \), adjacent side is \( 13 \), opposite side is \( x \). So \( \tan(32^\circ)=\frac{\text{opposite}}{\text{adjacent}}=\frac{x}{13} \).

Step2: Solve for \( x \)

Rearrange: \( x = 13\times\tan(32^\circ) \). \( \tan(32^\circ)\approx0.6249 \), so \( x = 13\times0.6249\approx8.1 \).

Step1: Identify trigonometric ratio

In right triangle \( \triangle ABC \), \( \angle C = 90^\circ \), \( \angle A = 50.1^\circ \), adjacent side is \( 5 \), opposite side is \( x \). So \( \tan(50.1^\circ)=\frac{\text{opposite}}{\text{adjacent}}=\frac{x}{5} \).

Step2: Solve for \( x \)

Rearrange: \( x = 5\times\tan(50.1^\circ) \). \( \tan(50.1^\circ)\approx1.199 \), so \( x = 5\times1.199\approx6.0 \) (more accurately, \( \tan(50.1^\circ)\approx1.1987 \), \( x = 5\times1.1987\approx6.0 \)).

Answer:

\( 13.8 \)

Problem 10