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. Use the identity \( \sin(\alpha) = \cos(90^\circ - \alpha) \) to rewrite the equation: \( \sin(2\theta + 10^\circ) = \cos(3\theta - 20^\circ) \) becomes \( \sin(2\theta + 10^\circ) = \sin(90^\circ - (3\theta - 20^\circ)) \).
2. Simplify the right side: \( \sin(90^\circ - (3\theta - 20^\circ)) = \sin(110^\circ - 3\theta) \).
3. Set the arguments of the sine functions equal to each other: \( 2\theta + 10^\circ = 110^\circ - 3\theta \).
4. Solve for \( \theta \) by first adding \( 3\theta \) to both sides: \( 2\theta + 3\theta + 10^\circ = 110^\circ \).
5. Combine like terms and solve for \( \theta \): \( 5\theta + 10^\circ = 110^\circ \), then subtract \( 10^\circ \) from both sides and divide by 5.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. A key identity relevant to this problem is the co-function identity, which states that sin(x) = cos(90° - x). This identity allows us to relate sine and cosine functions, facilitating the solving of equations that involve both.

Angle Addition Formulas

Angle addition formulas express the sine and cosine of the sum of two angles in terms of the sines and cosines of the individual angles. For example, sin(a + b) = sin(a)cos(b) + cos(a)sin(b). These formulas are essential for simplifying expressions like sin(2θ + 10°) and cos(3θ - 20°) in the given equation, making it easier to find solutions.

Acute Angles

Acute angles are angles that measure less than 90 degrees. In trigonometry, the values of sine and cosine for acute angles are always positive. This property is crucial when solving the equation, as it restricts the possible solutions to those that fall within the first quadrant, ensuring that the angles involved remain acute.