Textbook Question
Write each function as an expression involving functions of θ or x alone. See Example 2.
tan (π/4 + x)
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6. Trigonometric Identities and More Equations
Sum and Difference Identities
Problem 5.46
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(θ - 270°)
Verified step by step guidance1
Recognize that the expression \( \cos(\theta - 270^\circ) \) involves the cosine of a difference.
Recall the cosine difference identity: \( \cos(a - b) = \cos a \cos b + \sin a \sin b \).
Substitute \( a = \theta \) and \( b = 270^\circ \) into the identity: \( \cos(\theta - 270^\circ) = \cos \theta \cos 270^\circ + \sin \theta \sin 270^\circ \).
Evaluate \( \cos 270^\circ \) and \( \sin 270^\circ \): \( \cos 270^\circ = 0 \) and \( \sin 270^\circ = -1 \).
Substitute these values back into the expression: \( \cos(\theta - 270^\circ) = \cos \theta \cdot 0 + \sin \theta \cdot (-1) \).
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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 or Difference Identity
The cosine of a sum or difference identity states that cos(a ± b) = cos(a)cos(b) ∓ sin(a)sin(b). This identity allows us to express the cosine of an angle that is the sum or difference of two other angles in terms of the cosine and sine of those angles. In the given question, we will apply this identity to simplify cos(θ - 270°).
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Reference Angles
A reference angle is the acute angle formed by the terminal side of an angle and the x-axis. It is essential for evaluating trigonometric functions because it helps determine the sign and value of the function in different quadrants. For angles like -270°, understanding the reference angle can simplify calculations and lead to the correct evaluation of trigonometric functions.
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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, cosine, and tangent functions. By using the unit circle, we can easily find the values of trigonometric functions for various angles, including those expressed in degrees, such as -270°.
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