Find one value of θ or x that satisfies each of the following.
sin θ = cos(2θ + 30°)

Verified step by step guidance

1. Start by using the trigonometric identity for cosine of a double angle: \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \).
2. Rewrite the equation \( \sin \theta = \cos(2\theta + 30^\circ) \) using the identity: \( \cos(2\theta + 30^\circ) = \cos(2\theta)\cos(30^\circ) - \sin(2\theta)\sin(30^\circ) \).
3. Substitute the double angle identities: \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \) and \( \sin(30^\circ) = \frac{1}{2} \).
4. Set up the equation: \( \sin \theta = (\cos^2(\theta) - \sin^2(\theta))\frac{\sqrt{3}}{2} - 2\sin(\theta)\cos(\theta)\frac{1}{2} \).
5. Solve the equation for \( \theta \) by equating the expressions and simplifying.

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 question is the co-function identity, which states that sin(θ) = cos(90° - θ). This identity can help in transforming the equation sin θ = cos(2θ + 30°) into a more manageable form for solving.

Angle Addition Formulas

Angle addition formulas allow us to express the sine and cosine of sums or differences of angles in terms of the sine and cosine of the individual angles. For example, the cosine of a sum can be expressed as cos(A + B) = cosA cosB - sinA sinB. This concept is essential for expanding the right side of the equation cos(2θ + 30°) to facilitate solving for θ.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the angles that satisfy a given trigonometric relationship. This often requires isolating the trigonometric function and using inverse functions or identities to find solutions. In this case, after transforming the equation using identities, we can isolate θ and find its values within a specified range, typically 0° to 360°.