Textbook Question
Write each function value in terms of the cofunction of a complementary angle.
sin 98.0142°
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6. Trigonometric Identities and More Equations
Sum and Difference Identities
Problem 38
Textbook Question
Find one value of θ or x that satisfies each of the following.
sin θ = cos(2θ + 30°)
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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Period of Sine and Cosine Functions
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.
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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.
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