Using pythagorean identities show that 𝑐𝑜𝑠2𝜃 + 𝑠𝑖𝑛2𝜃 = 1

To prove that \( \cos^2\theta + \sin^2\theta = 1 \) using the Pythagorean identities, we'll start by recalling the fundamental identity related to the unit circle and trigonometric functions.

### Step 1: Recall the Unit Circle Definition
On the unit circle, any point \( P \) on the circle has coordinates \( (\cos\theta, \sin\theta) \) where \( \theta \) is the angle formed with the positive x-axis.

### Step 2: Apply the Pythagorean Theorem
The equation of a circle with radius \( r \) centered at the origin is:
\[
x^2 + y^2 = r^2
\]
For the unit circle, \( r = 1 \), so:
\[
x^2 + y^2 = 1
\]
Since \( x = \cos\theta \) and \( y = \sin\theta \), substituting these into the equation gives:
\[
\cos^2\theta + \sin^2\theta = 1
\]

### Conclusion:
Thus, we've shown that:
\[
\cos^2\theta + \sin^2\theta = 1
\]
which is one of the fundamental Pythagorean identities in trigonometry.

Would you like more details or have any questions?

---

Here are five related questions:

1. How can the identity \( \cos^2\theta + \sin^2\theta = 1 \) be used to derive other trigonometric identities?
2. What is the geometric interpretation of the Pythagorean identity on the unit circle?
3. How can you use \( \cos^2\theta + \sin^2\theta = 1 \) to find the sine or cosine of an angle if the other is known?
4. What are the other Pythagorean identities in trigonometry?
5. How does the identity \( \cos^2\theta + \sin^2\theta = 1 \) relate to the equation of a circle in Cartesian coordinates?

**Tip:** Remember that the Pythagorean identities are fundamental in simplifying trigonometric expressions and solving trigonometric equations.

Learn how to prove the Pythagorean identity \( \cos^2\theta + \sin^2\theta = 1 \) using the unit circle and trigonometric functions. This detailed solution covers fundamental trigonometric concepts and is suitable for 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