Question

a triangle has sides with lengths 5,12, and 13.

1. approximate the acute angles in this triangle.

a. the measure of the smaller angle = ______ degrees
b. the measure of the larger angel = ______ degrees

Explanation:

Step1: Use trigonometric ratios

For a right - triangle with sides \(a = 5\), \(b=12\), and \(c = 13\) (where \(c\) is the hypotenuse). Let \(\theta_1\) be the angle opposite to side \(a = 5\) and \(\theta_2\) be the angle opposite to side \(b = 12\).
We know that \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\) and \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), and also \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Using \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), for the smaller angle \(\theta_1\) (opposite side \(a = 5\), adjacent side \(b = 12\)), \(\tan\theta_1=\frac{5}{12}\).

Step2: Calculate the smaller angle

We use the inverse - tangent function \(\theta=\tan^{- 1}(x)\). If \(\tan\theta_1=\frac{5}{12}\), then \(\theta_1=\tan^{-1}(\frac{5}{12})\).
Using a calculator, \(\theta_1=\tan^{-1}(\frac{5}{12})\approx22.6^{\circ}\) (since \(\tan^{-1}(0.4167)\approx22.6^{\circ}\)).

Step3: Calculate the larger angle

Since the two acute angles in a right - triangle are complementary (\(\theta_1+\theta_2 = 90^{\circ}\)).
If \(\theta_1\approx22.6^{\circ}\), then \(\theta_2=90^{\circ}-\theta_1\).
\(\theta_2 = 90^{\circ}-22.6^{\circ}=67.4^{\circ}\)

Answer:

a. \(22.6\) degrees
b. \(67.4\) degrees