https://r2.erweima.ai/mathbot/image/1726443454797qhuwpvqz.jpg

The image shows a right triangle \( \triangle ABC \) with the following sides:

- \( AC = \sqrt{2} \)
- \( BC = 2\sqrt{2} \)
- \( AB = c \) (the hypotenuse)

You are asked to find the trigonometric ratios for angle \( A \), specifically:

1. \( \sin A \)
2. \( \cos A \)
3. \( \tan A \)

### Step 1: Hypotenuse \( c \)
We can calculate the hypotenuse \( AB = c \) using the Pythagorean theorem:
\[
c^2 = AC^2 + BC^2
\]
\[
c^2 = (\sqrt{2})^2 + (2\sqrt{2})^2 = 2 + 8 = 10
\]
\[
c = \sqrt{10}
\]

### Step 2: Trigonometric Ratios
For angle \( A \):
1. \( \sin A = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AB} = \frac{2\sqrt{2}}{\sqrt{10}} \)
2. \( \cos A = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AC}{AB} = \frac{\sqrt{2}}{\sqrt{10}} \)
3. \( \tan A = \frac{\text{opposite}}{\text{adjacent}} = \frac{BC}{AC} = \frac{2\sqrt{2}}{\sqrt{2}} = 2 \)

### Simplified results:
1. \( \sin A = \frac{2\sqrt{2}}{\sqrt{10}} \)
2. \( \cos A = \frac{\sqrt{2}}{\sqrt{10}} \)
3. \( \tan A = 2 \)

Let me know if you need further explanation!

### Related Questions:
1. How do you simplify expressions involving square roots?
2. Can you compute the angles of the triangle using trigonometric inverses?
3. How does the Pythagorean theorem work in non-right triangles?
4. What are the exact values for \( \sin A \) and \( \cos A \) after rationalizing the denominators?
5. What is the importance of knowing the exact trigonometric ratios in solving real-life problems?

### Tip:
Remember to always rationalize the denominator when working with square roots in trigonometric functions for cleaner results!

Calculate the trigonometric ratios (SinA, CosA, TanA) for a right triangle where AC = √2, BC = 2√2, and AB is the hypotenuse.

This problem involves finding the trigonometric ratios (sin, cos, tan) for a right triangle with side lengths AC = √2, BC = 2√2, and AB as the hypotenuse. Learn to apply the Pythagorean Theorem and trigonometric formulas to calculate the exact ratios for SinA, CosA, and TanA.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation