prove the identify sin squared x + cos squared x =1 using the pythagorean therom and a unit circle diagram

To prove the identity \( \sin^2(x) + \cos^2(x) = 1 \) using the Pythagorean theorem and a unit circle diagram, follow these steps:

### 1. **Understanding the Unit Circle:**
   - The unit circle is a circle centered at the origin \((0, 0)\) in the coordinate plane, with a radius of 1.
   - Any point on the unit circle can be represented as \((\cos(x), \sin(x))\), where \(x\) is the angle formed with the positive x-axis.

### 2. **Visualize a Point on the Unit Circle:**
   - Consider a point \(P\) on the unit circle corresponding to the angle \(x\). The coordinates of this point are \( (\cos(x), \sin(x)) \).
   - Draw a right triangle by dropping a perpendicular from point \(P\) to the x-axis. This forms a right triangle with:
     - Hypotenuse: The radius of the circle, which is \(1\).
     - Adjacent side (along the x-axis): \( \cos(x) \).
     - Opposite side (along the y-axis): \( \sin(x) \).

### 3. **Apply the Pythagorean Theorem:**
   - The Pythagorean theorem states that in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse:
     \[
     \text{(Adjacent side)}^2 + \text{(Opposite side)}^2 = \text{(Hypotenuse)}^2
     \]
   - Substituting the values from the triangle:
     \[
     \cos^2(x) + \sin^2(x) = 1^2
     \]
   - Simplifying this, we get:
     \[
     \cos^2(x) + \sin^2(x) = 1
     \]
   - This equation is the Pythagorean identity, which is true for any angle \(x\).

### 4. **Conclusion:**
   - The identity \( \sin^2(x) + \cos^2(x) = 1 \) is directly derived from the Pythagorean theorem applied to the right triangle formed by a point on the unit circle.

Would you like to see a detailed unit circle diagram with the explanation? If you have any questions or want further clarification, feel free to ask!

Here are 5 related questions you might find interesting:
1. How can the identity \( \sin^2(x) + \cos^2(x) = 1 \) be used in solving trigonometric equations?
2. What are the other fundamental trigonometric identities related to the unit circle?
3. How does the identity change if the circle is not a unit circle?
4. How can this identity be applied in physics, particularly in wave motion?
5. What are the implications of this identity in vector mathematics?

**Tip:** Always remember that the unit circle is a powerful tool for understanding trigonometric functions and identities because it visually represents the relationship between angles and side lengths in right triangles.

Learn how to prove the trigonometric identity sin^2(x) + cos^2(x) = 1 using the Pythagorean theorem and a unit circle diagram. This problem explores fundamental trigonometric concepts and the application of the Pythagorean identity. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation