Textbook Question
Write each function as an expression involving functions of θ or x alone. See Example 2.
tan(180° + θ)
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6. Trigonometric Identities and More Equations
Sum and Difference Identities
Problem 5.50
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
Recognize that the expression involves the cosine of a sum: \( \cos(270^\circ + \theta) \).
Use the cosine of a sum identity: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \).
Substitute \( a = 270^\circ \) and \( b = \theta \) into the identity: \( \cos(270^\circ + \theta) = \cos 270^\circ \cos \theta - \sin 270^\circ \sin \theta \).
Recall the exact values: \( \cos 270^\circ = 0 \) and \( \sin 270^\circ = -1 \).
Substitute these values into the expression: \( \cos(270^\circ + \theta) = 0 \cdot \cos \theta - (-1) \cdot \sin \theta \).
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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 identity allows us to express the cosine of the sum of two angles in terms of the cosines and sines of those angles, which is essential for simplifying expressions involving angle sums.
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Reference Angles
Reference angles are the acute angles formed by the terminal side of an angle and the x-axis. For angles greater than 180°, such as 270°, understanding the reference angle helps in determining the cosine and sine values, as these values can be derived from the corresponding acute angle in the first quadrant.
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Unit Circle
The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is a fundamental tool in trigonometry, as it provides a geometric interpretation of sine and cosine values for any angle, allowing for easy calculation of these functions based on the angle's position on the circle.
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