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(90° + θ)
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6. Trigonometric Identities and More Equations
Sum and Difference Identities
Problem 16
Textbook Question
Find the exact value of each expression. (Do not use a calculator.)
cos (-7π/12)
Verified step by step guidance1
Recall the even-odd property of cosine: \(\cos(-\theta) = \cos(\theta)\). So, \(\cos\left(-\frac{7\pi}{12}\right) = \cos\left(\frac{7\pi}{12}\right)\).
Express \(\frac{7\pi}{12}\) as a sum or difference of angles whose cosine and sine values are known. For example, \(\frac{7\pi}{12} = \frac{3\pi}{4} - \frac{\pi}{6}\).
Use the cosine difference identity: \(\cos(a - b) = \cos a \cos b + \sin a \sin b\).
Substitute \(a = \frac{3\pi}{4}\) and \(b = \frac{\pi}{6}\) into the identity: \(\cos\left(\frac{3\pi}{4} - \frac{\pi}{6}\right) = \cos\frac{3\pi}{4} \cos\frac{\pi}{6} + \sin\frac{3\pi}{4} \sin\frac{\pi}{6}\).
Recall the exact values: \(\cos\frac{3\pi}{4} = -\frac{\sqrt{2}}{2}\), \(\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}\), \(\sin\frac{3\pi}{4} = \frac{\sqrt{2}}{2}\), and \(\sin\frac{\pi}{6} = \frac{1}{2}\). Substitute these to write the expression fully.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Unit Circle and Angle Measurement
The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles are measured in radians, where 2π radians equal 360 degrees. Understanding how to locate angles on the unit circle, including negative angles which represent clockwise rotation, is essential for evaluating trigonometric functions like cosine.
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Cosine Function and Even-Odd Properties
Cosine is an even function, meaning cos(-θ) = cos(θ). This property allows simplification of expressions with negative angles by converting them to positive angles, making it easier to find exact values using known reference angles on the unit circle.
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Angle Sum and Difference Identities
The cosine of a sum or difference of angles can be expressed using identities: cos(a ± b) = cos a cos b ∓ sin a sin b. These identities help break down complex angles like 7π/12 into sums or differences of standard angles (e.g., π/3 and π/4) whose sine and cosine values are known exactly.
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