Question

there are two right triangles. the left triangle xyz has a right angle at x, with xz = 18.4, xy = 34.5, and yz = 39.1. the right triangle abc has a right angle at a. we need to find sinb, tanb, and cosb.

Explanation:

Step1: Identify Similar Triangles

Triangles \( \triangle XYZ \) and \( \triangle ABC \) are right - angled ( \( \angle X = \angle A=90^{\circ} \)) and similar. So the ratios of corresponding sides are equal. In \( \triangle XYZ \), \( XZ = 18.4 \), \( XY = 34.5 \), \( YZ = 39.1 \). In \( \triangle ABC \), for angle \( B \), the opposite side to \( B \) is \( AC \) (corresponding to \( XZ \)), the adjacent side is \( AB \) (corresponding to \( XY \)), and the hypotenuse is \( BC \) (corresponding to \( YZ \)).

Step2: Calculate \( \sin B \)

The formula for sine of an angle in a right - triangle is \( \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}} \). For angle \( B \), opposite side \( AC = XZ = 18.4 \), hypotenuse \( BC=YZ = 39.1 \). So \( \sin B=\frac{18.4}{39.1}\approx0.4706 \)

Step3: Calculate \( \tan B \)

The formula for tangent of an angle in a right - triangle is \( \tan\theta=\frac{\text{opposite}}{\text{adjacent}} \). For angle \( B \), opposite side \( AC = 18.4 \), adjacent side \( AB = 34.5 \). So \( \tan B=\frac{18.4}{34.5}\approx0.5333 \)

Step4: Calculate \( \cos B \)

The formula for cosine of an angle in a right - triangle is \( \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}} \). For angle \( B \), adjacent side \( AB = 34.5 \), hypotenuse \( BC = 39.1 \). So \( \cos B=\frac{34.5}{39.1}\approx0.8824 \)

Answer:

\( \sin B\approx\boldsymbol{0.471} \), \( \tan B\approx\boldsymbol{0.533} \), \( \cos B\approx\boldsymbol{0.882} \) (rounded to three decimal places)