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Ch. 3 - Trigonometric Identities and Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblem 7b
Chapter 3, Problem 7b

Each expression is the right side of the formula for cos (α - β) with particular values for α and β. Write the expression as the cosine of an angle.
cos5π12cosπ12+sin5π12sinπ12\(\cos\) \(\frac{5\pi}{12}\) \(\cos\) \(\frac{\pi}{12}\) + \(\sin\) \(\frac{5\pi}{12}\) \(\sin\) \(\frac{\pi}{12}\)

Verified step by step guidance
1
Identify the given expression as matching the cosine difference formula: \(\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta\).
Compare the given expression \(\cos \frac{5\pi}{12} \cos \frac{\pi}{12} + \sin \frac{5\pi}{12} \sin \frac{\pi}{12}\) to the formula and recognize that \(\alpha = \frac{5\pi}{12}\) and \(\beta = \frac{\pi}{12}\).
Apply the formula by rewriting the expression as \(\cos \left( \frac{5\pi}{12} - \frac{\pi}{12} \right)\).
Simplify the angle inside the cosine function by subtracting the fractions: \(\frac{5\pi}{12} - \frac{\pi}{12} = \frac{4\pi}{12}\).
Reduce the fraction \(\frac{4\pi}{12}\) to its simplest form \(\frac{\pi}{3}\), so the expression becomes \(\cos \frac{\pi}{3}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Cosine of a Difference Formula

The cosine of a difference of two angles, α and β, is given by cos(α - β) = cos α cos β + sin α sin β. This identity allows rewriting expressions involving products of sines and cosines as a single cosine function of the difference of angles.
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Verifying Identities with Sum and Difference Formulas

Angle Simplification and Substitution

To rewrite the given expression as a single cosine, identify α and β from the terms. This involves recognizing the angles in radians and substituting them into the formula, then simplifying the difference α - β to find the equivalent angle.
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Solve Trig Equations Using Identity Substitutions

Radian Measure and Angle Arithmetic

Understanding radian measure is essential for manipulating angles like 5π/12 and π/12. Adding or subtracting these fractions requires common denominators and simplification to find the exact angle represented by α - β.
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Converting between Degrees & Radians
Related Practice
Textbook Question
In Exercises 7–14, use the given information to find the exact value of each of the following:b. cos 2θ15sin θ = -------- , θ lies in quadrant II.17
Textbook Question
In Exercises 7–14, use the given information to find the exact value of each of the following:a. sin 2θ15sin θ = -------- , θ lies in quadrant II.17
Textbook Question

Use the given information to find the exact value of each of the following: sin2θ\(\sin\)2\(\theta\)

sinθ=1213,θ lies in quadrant II.\(\sin\) \(\theta\) = \(\frac{12}{13}\), \(\quad\) \(\theta\) \(\text{ lies in quadrant II.}\)

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Textbook Question

Each expression is the right side of the formula for cos (α - β) with particular values for α and β. Find the exact value of the expression.

cos5π12cosπ12+sin5π12sinπ12\(\cos\) \(\frac{5\pi}{12}\) \(\cos\) \(\frac{\pi}{12}\) + \(\sin\) \(\frac{5\pi}{12}\) \(\sin\) \(\frac{\pi}{12}\)

Textbook Question

Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference.

cos3x2sinx2\(\cos\) \(\frac{3x}{2}\) \(\sin\) \(\frac{x}{2}\)

Textbook Question

In Exercises 7–14, use the given information to find the exact value of each of the following: c. tan 2θ 15 sin θ = -------- , θ lies in quadrant II. 17

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