https://file.aiquickdraw.com/mathbot/file/1725499077413v4pjknqv.jpg

Let's analyze the triangle in the image and calculate the trigonometric values requested (cosine, sine, and tangent for angle \(A\)).

### Step 1: Identify the triangle's dimensions
- From the figure:
  - The triangle on the left has sides:
    - \( XY = 19.5 \)
    - \( YZ = 22.1 \)
    - \( XZ = 30.4 \)

This triangle seems to be used to find trigonometric ratios for angle \(A\).

### Step 2: Use trigonometric identities
For a right triangle, the trigonometric ratios are defined as:
- \( \cos(A) = \frac{\text{adjacent}}{\text{hypotenuse}} \)
- \( \sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} \)
- \( \tan(A) = \frac{\text{opposite}}{\text{adjacent}} \)

Based on the information:
- The hypotenuse of triangle \(XYZ\) is \( XZ = 30.4 \).
- The side adjacent to angle \( A \) (in triangle \( XYZ \)) is \( XY = 19.5 \).
- The side opposite to angle \( A \) is \( YZ = 22.1 \).

Now, let me calculate the values of \( \cos(A) \), \( \sin(A) \), and \( \tan(A) \).# Given side lengths from the triangle
adjacent = 19.5  # side XY (adjacent to angle A)
opposite = 22.1  # side YZ (opposite to angle A)
hypotenuse = 30.4  # side XZ (hypotenuse)

# Calculating trigonometric ratios
import math

cos_A = adjacent / hypotenuse  # Cosine of angle A
sin_A = opposite / hypotenuse  # Sine of angle A
tan_A = opposite / adjacent    # Tangent of angle A

# Rounding the answers to the nearest hundredth
cos_A_rounded = round(cos_A, 2)
sin_A_rounded = round(sin_A, 2)
tan_A_rounded = round(tan_A, 2)

cos_A_rounded, sin_A_rounded, tan_A_roundedThe trigonometric values for angle \( A \) are:

- \( \cos(A) = 0.64 \)
- \( \sin(A) = 0.73 \)
- \( \tan(A) = 1.13 \)

These values are rounded to the nearest hundredth.

Would you like more details or have any questions? Here are some follow-up questions to explore:

1. How are trigonometric ratios used in different areas of mathematics?
2. What is the relationship between sine, cosine, and tangent in a right triangle?
3. How do you find the angles in a triangle if the sides are given?
4. What are the applications of trigonometry in real-world problems?
5. How do you calculate the missing side of a right triangle using trigonometric ratios?

**Tip:** Remember that in right triangles, the hypotenuse is always the longest side, and its length influences the sine and cosine of the angles.

In the figure below, find cosA, sinA, and tanA for angle A in triangle XYZ. Round your answers to the nearest hundredth.

This problem demonstrates how to calculate the trigonometric ratios (cosine, sine, and tangent) for angle A in a right triangle. The given sides of the triangle are used to find cos(A), sin(A), and tan(A), rounded to the nearest hundredth. Suitable for students in Grades 9-11.

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