Textbook Question
Write each function as an expression involving functions of θ or x alone. See Example 2.
sin (3π/4 - x)
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6. Trigonometric Identities and More Equations
Sum and Difference Identities
Problem 50
Textbook Question
Use the identities for the cosine of a sum or difference to write each expression as a trigonometric function of θ alone.
cos(270° + θ)
Verified step by step guidance1
Recall the cosine sum identity: \(\cos(A + B) = \cos A \cos B - \sin A \sin B\).
Identify \(A = 270^\circ\) and \(B = \theta\) in the expression \(\cos(270^\circ + \theta)\).
Apply the identity: \(\cos(270^\circ + \theta) = \cos 270^\circ \cos \theta - \sin 270^\circ \sin \theta\).
Use known exact values: \(\cos 270^\circ = 0\) and \(\sin 270^\circ = -1\).
Substitute these values back into the expression to write it solely in terms of \(\theta\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Cosine of a Sum Identity
The cosine of a sum identity states that cos(A + B) = cos A cos B - sin A sin B. This formula allows you to express the cosine of the sum of two angles in terms of the cosines and sines of the individual angles, which is essential for rewriting expressions like cos(270° + θ).
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Special Angle Values
Special angles such as 0°, 90°, 180°, 270°, and 360° have known sine and cosine values. For example, cos 270° = 0 and sin 270° = -1. Using these values simplifies expressions involving these angles, enabling the reduction of cos(270° + θ) to a function involving only θ.
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Trigonometric Function Simplification
After applying identities and substituting known values, simplifying the resulting expression involves combining like terms and recognizing standard trigonometric functions. This step ensures the final expression is written solely in terms of θ, as required.
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