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. Recall the co-function identity in trigonometry: \(\sin x = \cos(90^\circ - x)\). Use this to rewrite the right side of the equation \(\cos(2\theta + 50^\circ) = \sin(2\theta - 20^\circ)\) as \(\cos(90^\circ - (2\theta - 20^\circ))\).
2. Simplify the expression inside the cosine on the right side: \(90^\circ - (2\theta - 20^\circ) = 90^\circ - 2\theta + 20^\circ = 110^\circ - 2\theta\).
3. Now the equation becomes \(\cos(2\theta + 50^\circ) = \cos(110^\circ - 2\theta)\). Since the cosines of two angles are equal, set the angles equal to each other or their supplements: either \(2\theta + 50^\circ = 110^\circ - 2\theta\) or \(2\theta + 50^\circ = 360^\circ - (110^\circ - 2\theta)\).
4. Solve the first equation \(2\theta + 50^\circ = 110^\circ - 2\theta\) for \(\theta\) by isolating \(\theta\) on one side.
5. Check the solution to ensure \(\theta\) is an acute angle (between \(0^\circ\) and \(90^\circ\)). If it is, this is a valid solution. If not, solve the second equation and check again.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. In this problem, recognizing that sin(x) = cos(90° - x) allows us to rewrite and equate angles, facilitating the solution of the equation.

Solving Trigonometric Equations

Solving trigonometric equations involves manipulating the equation using identities and algebraic techniques to isolate the variable. Since the problem restricts angles to acute values, solutions must be checked within the interval (0°, 90°) to ensure validity.

Angle Restrictions and Domain Considerations

The problem specifies that all angles are acute, meaning between 0° and 90°. This restriction limits possible solutions and helps avoid extraneous answers that arise from the periodic nature of trigonometric functions.