Find one solution for each equation. Assume all angles involved are acute angles. See Example 3.cos(2θ + 50°) = sin(2θ - 20°)

Verified step by step guidance

1. Use the identity \( \cos A = \sin(90^\circ - A) \) to rewrite the equation: \( \cos(2\theta + 50^\circ) = \sin(2\theta - 20^\circ) \) becomes \( \cos(2\theta + 50^\circ) = \cos(90^\circ - (2\theta - 20^\circ)) \).
2. Simplify the right side: \( \cos(2\theta + 50^\circ) = \cos(110^\circ - 2\theta) \).
3. Since \( \cos A = \cos B \) implies \( A = B + 360^\circ k \) or \( A = -B + 360^\circ k \) for integer \( k \), set \( 2\theta + 50^\circ = 110^\circ - 2\theta + 360^\circ k \).
4. Solve for \( \theta \) in the equation \( 2\theta + 50^\circ = 110^\circ - 2\theta + 360^\circ k \).
5. Rearrange and simplify to find \( \theta \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Key identities include the Pythagorean identity, angle sum and difference identities, and co-function identities. Understanding these identities is crucial for transforming and simplifying trigonometric equations, such as converting sine to cosine or vice versa.

Angle Relationships

In trigonometry, angle relationships refer to how different angles relate to each other, particularly in terms of complementary and supplementary angles. For acute angles, the sine of an angle is equal to the cosine of its complement. This relationship is essential for solving equations where sine and cosine functions are involved, as it allows for the substitution of one function for another.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the angles that satisfy a given trigonometric equation. This process often requires using identities to rewrite the equation in a more manageable form, isolating the trigonometric function, and then determining the angle solutions. For acute angles, solutions must be within the range of 0° to 90°, which is important for ensuring the validity of the solutions.