Textbook Question
Find the exact value of each expression. (Do not use a calculator.)
cos(-15°)
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6. Trigonometric Identities and More Equations
Sum and Difference Identities
Problem 18
Textbook Question
Find the exact value of each expression. (Do not use a calculator.)
cos (7π/9) cos (2π/9) - sin (7π/9) sin (2π/9)
Verified step by step guidance1
Recognize that the expression matches the cosine addition formula: \(\cos A \cos B - \sin A \sin B = \cos(A + B)\).
Identify the angles in the expression: \(A = \frac{7\pi}{9}\) and \(B = \frac{2\pi}{9}\).
Apply the formula by substituting the values of \(A\) and \(B\): \(\cos\left(\frac{7\pi}{9}\right) \cos\left(\frac{2\pi}{9}\right) - \sin\left(\frac{7\pi}{9}\right) \sin\left(\frac{2\pi}{9}\right) = \cos\left(\frac{7\pi}{9} + \frac{2\pi}{9}\right)\).
Add the angles inside the cosine function: \(\frac{7\pi}{9} + \frac{2\pi}{9} = \frac{9\pi}{9} = \pi\).
Use the known exact value of \(\cos \pi\) to find the final exact value.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Cosine of Sum and Difference Identities
These identities express the cosine of a sum or difference of two angles in terms of the sines and cosines of the individual angles. Specifically, cos(A + B) = cos A cos B - sin A sin B, which matches the given expression and allows simplification without a calculator.
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Exact Values of Trigonometric Functions at Special Angles
Certain angles, especially those related to π/3, π/4, π/6, and their multiples, have known exact sine and cosine values. Recognizing or converting angles to these special angles helps in finding exact trigonometric values without approximation.
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Angle Measurement in Radians
Radians measure angles based on the radius of a circle, where 2π radians equal 360 degrees. Understanding radian measure is essential for interpreting and manipulating trigonometric expressions involving π, such as 7π/9 and 2π/9.
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