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

검증된 단계별 안내

Recall the co-function identity in trigonometry: \(\sin \theta = \cos(90^\circ - \theta)\). This means we can rewrite the equation \(\sin \theta = \cos(2\theta + 30^\circ)\) as \(\sin \theta = \cos(90^\circ - \theta)\) to find a relationship between \(\theta\) and \(2\theta + 30^\circ\).

Set the angles inside the cosine equal to each other using the identity: \(2\theta + 30^\circ = 90^\circ - \theta\). This gives an equation involving \(\theta\) that you can solve.

Solve the linear equation \(2\theta + 30^\circ = 90^\circ - \theta\) by first adding \(\theta\) to both sides and subtracting \(30^\circ\) from both sides to isolate \(\theta\) terms on one side.

Simplify the resulting equation to find the value of \(\theta\). This will give you one possible solution for \(\theta\) that satisfies the original equation.

Remember that sine and cosine are periodic functions, so there may be multiple solutions. After finding one solution, consider the general solutions by adding the appropriate multiples of \(360^\circ\) or \(2\pi\) radians if needed.

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주요 개념

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Relationship Between Sine and Cosine Functions

Sine and cosine are co-function trigonometric functions related by phase shifts. Specifically, sin(θ) = cos(90° - θ), which allows transforming equations involving sine and cosine into a single trigonometric function for easier solving.

Solving Trigonometric Equations

Solving equations like sin θ = cos(2θ + 30°) involves using identities and algebraic manipulation to isolate θ. This often requires rewriting one function in terms of the other and considering the periodic nature of trigonometric functions to find all possible solutions.

Angle Measures and Degree Conversion

Understanding angle measures in degrees and how to manipulate them is essential. When dealing with expressions like 2θ + 30°, it’s important to correctly apply arithmetic operations and consider the periodicity of trigonometric functions in degrees to find valid solutions.

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