In Exercises 63–68, find the exact value of each expression. Do not use a calculator.1 + sin² 40° + sin² 50°

Verified step by step guidance

1. Recognize that the expression involves trigonometric identities, specifically the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \).
2. Notice that \( \sin^2 40^\circ + \sin^2 50^\circ \) can be rewritten using the identity \( \sin^2 \theta = 1 - \cos^2 \theta \).
3. Substitute \( \sin^2 40^\circ = 1 - \cos^2 40^\circ \) and \( \sin^2 50^\circ = 1 - \cos^2 50^\circ \) into the expression.
4. Simplify the expression to \( 1 + (1 - \cos^2 40^\circ) + (1 - \cos^2 50^\circ) \).
5. Combine like terms to further simplify the expression.

Key Concepts

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. This fundamental relationship between sine and cosine is crucial for simplifying trigonometric expressions and solving equations. Understanding this identity allows students to manipulate and combine sine and cosine functions effectively.

Sine Function Properties

The sine function is periodic and has specific values for common angles. Notably, sin(40°) and sin(50°) can be related through the complementary angle identity, where sin(θ) = cos(90° - θ). Recognizing these properties helps in evaluating expressions involving sine functions without a calculator.

Sum of Sine Squares

The expression sin²(θ) + sin²(φ) can often be simplified using trigonometric identities. In this case, knowing that sin²(θ) + sin²(φ) can be expressed in terms of cosines or combined using angle addition formulas is essential. This understanding aids in finding exact values for trigonometric expressions involving multiple angles.