Find one solution for each equation. Assume all angles involved are acute angles. See Example 3. sin(2θ + 10°) = cos(3θ - 20°)

Verified step by step guidance

1. Recall the co-function identity: for any angle \( x \), \( \sin x = \cos(90^\circ - x) \). Use this to rewrite the equation \( \sin(2\theta + 10^\circ) = \cos(3\theta - 20^\circ) \) as \( \sin(2\theta + 10^\circ) = \sin(90^\circ - (3\theta - 20^\circ)) \).
2. Simplify the right side inside the sine function: \( 90^\circ - (3\theta - 20^\circ) = 90^\circ - 3\theta + 20^\circ = 110^\circ - 3\theta \). So the equation becomes \( \sin(2\theta + 10^\circ) = \sin(110^\circ - 3\theta) \).
3. Use the sine equation property: if \( \sin A = \sin B \), then either \( A = B + 360^\circ k \) or \( A = 180^\circ - B + 360^\circ k \), where \( k \) is any integer. Since we are looking for acute angles, consider \( k = 0 \) and write the two possible equations:
4. \( 2\theta + 10^\circ = 110^\circ - 3\theta \) and \( 2\theta + 10^\circ = 180^\circ - (110^\circ - 3\theta) \).
5. Solve each equation for \( \theta \) separately, then check which solution(s) fall within the range of acute angles (between 0° and 90°).

Key Concepts

Relationship Between Sine and Cosine Functions

Sine and cosine functions are co-functions, meaning sin(α) = cos(90° - α). This identity allows us to rewrite equations involving sine and cosine in terms of a single trigonometric function, facilitating the solving of equations like sin(2θ + 10°) = cos(3θ - 20°).

Solving Trigonometric Equations

Solving trigonometric equations involves using identities and algebraic manipulation to isolate the variable. For equations with multiple angles, it is important to consider the domain restrictions and use inverse trigonometric functions to find possible angle values.

Acute Angle Assumption and Domain Restrictions

Since the problem specifies all angles are acute, solutions must lie between 0° and 90°. This restriction limits the possible values of θ and helps in selecting the correct solution from multiple candidates obtained when solving the equation.