Find m<X ### Geometry Exercise: Finding an Angle in a TriangleIn the diagram, we have a triangle XYZ. The sides of the triangle are labeled as follows:- Side XY = 5 units- Side YZ = 12 units- Side XZ = 13 unitsPoint Y has a right angle (indicated by a small square). This means triangle XYZ is a right triangle. Given this information, you are asked to find the measure of angle ∠X.The options provided are:- 21.04°- 22.62°- 67.38°- 68.96°### Detailed Explanation:1. **Right Triangle Properties:** - In a right triangle, the side opposite the right angle (90°) is the hypotenuse. In this triangle, XZ is the hypotenuse with a length of 13 units. - The other two sides are known as the legs. Here, XY = 5 units and YZ = 12 units.2. **Using Trigonometric Ratios:** - To determine angle ∠X, we can use the trigonometric functions sine (sin), cosine (cos), or tangent (tan). For right triangle calculations, these functions relate the angles to the lengths of the sides: - sine: $\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}$ - cosine: $\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}$ - tangent: $\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}$3. **Calculation for Angle ∠X:** - To find angle ∠X at vertex X, we use the legs XY and YZ since they are known: - $\tan(\angle X) = \frac{YZ}{XY} = \frac{12}{5}$ - Finding the angle: - $\angle X = \tan^{-1}(\frac{12}{5})$4. **Using a Calculator:** - Use a scientific calculator or an online calculator to find the inverse tangent value (tan^-1). - $\angle X \approx 67.38°$Therefore, the correct answer for the measure of angle ∠X is:- **67

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Math

Trigonometry

Find m

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section: Chapter Questions

Problem 75RE

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Question

Find m<X

### Geometry Exercise: Finding an Angle in a Triangle

In the diagram, we have a triangle XYZ. The sides of the triangle are labeled as follows:
- Side XY = 5 units
- Side YZ = 12 units
- Side XZ = 13 units

Point Y has a right angle (indicated by a small square). This means triangle XYZ is a right triangle. Given this information, you are asked to find the measure of angle ∠X.

The options provided are:
- 21.04°
- 22.62°
- 67.38°
- 68.96°

### Detailed Explanation:

1. **Right Triangle Properties:**
- In a right triangle, the side opposite the right angle (90°) is the hypotenuse. In this triangle, XZ is the hypotenuse with a length of 13 units.
- The other two sides are known as the legs. Here, XY = 5 units and YZ = 12 units.

2. **Using Trigonometric Ratios:**
- To determine angle ∠X, we can use the trigonometric functions sine (sin), cosine (cos), or tangent (tan). For right triangle calculations, these functions relate the angles to the lengths of the sides:
- sine: $\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}$
- cosine: $\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}$
- tangent: $\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}$

3. **Calculation for Angle ∠X:**
- To find angle ∠X at vertex X, we use the legs XY and YZ since they are known:
- $\tan(\angle X) = \frac{YZ}{XY} = \frac{12}{5}$

- Finding the angle:
- $\angle X = \tan^{-1}(\frac{12}{5})$

4. **Using a Calculator:**
- Use a scientific calculator or an online calculator to find the inverse tangent value (tan^-1).
- $\angle X \approx 67.38°$

Therefore, the correct answer for the measure of angle ∠X is:

- **67

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