https://file.aiquickdraw.com/mathbot/file/1728315157890ynt4wlji.jpg

The problem involves a right-angled triangle \( XYZ \), with \( XY = 5 \, \text{cm} \) and \( XZ = 13 \, \text{cm} \). The goal is to find the values for the expressions given:

1. \( \text{A: } \tan X + \tan Z \)
2. \( \text{B: } \cos X \cos Z - \sin X \sin Z \)
3. \( \text{C: } \sin X \cos Z + \cos X \sin Z \)

### Step 1: Identify triangle sides
Since this is a right-angled triangle, we can use the Pythagorean theorem to find the length of the third side \( YZ \):

\[
YZ = \sqrt{XZ^2 - XY^2} = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 \, \text{cm}.
\]

### Step 2: Calculate the trigonometric values
- For \( \angle X \):
  \[
  \sin X = \frac{XY}{XZ} = \frac{5}{13}, \quad \cos X = \frac{YZ}{XZ} = \frac{12}{13}.
  \]
  
- For \( \angle Z \) (since \( Z \) is complementary to \( X \), \( \sin Z = \cos X \) and \( \cos Z = \sin X \)):
  \[
  \sin Z = \frac{12}{13}, \quad \cos Z = \frac{5}{13}.
  \]

### Step 3: Evaluate the given expressions
1. **A: \( \tan X + \tan Z \)**
   \[
   \tan X = \frac{\sin X}{\cos X} = \frac{5}{12}, \quad \tan Z = \frac{\sin Z}{\cos Z} = \frac{12}{5}.
   \]
   So,
   \[
   \tan X + \tan Z = \frac{5}{12} + \frac{12}{5} = \frac{25 + 144}{60} = \frac{169}{60}.
   \]

2. **B: \( \cos X \cos Z - \sin X \sin Z \)** (this is the formula for \( \cos(X + Z) \), and since \( X + Z = 90^\circ \), \( \cos 90^\circ = 0 \)):
   \[
   \cos X \cos Z - \sin X \sin Z = 0.


XYZ est un triangle rectangle en y tel que XY = 5 cm et XZ = 13 cm. Trouve la valeur de: tg X + tg Z, cos X cos Z − sin X sin Z, sin X cos Z + cos X sin Z.

Find the values of tangent, sine, and cosine for a right triangle XYZ with given sides XY = 5 cm and XZ = 13 cm. Learn how to apply the Pythagorean Theorem, trigonometric functions, and angle sum identities in this problem, suitable for Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation