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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 7c
Chapter 3, Problem 7c

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

Verified step by step guidance
1
Identify the given expression as the right side of the cosine difference formula: \(\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta\).
Match the angles in the expression to \(\alpha\) and \(\beta\). Here, \(\alpha = \frac{5\pi}{12}\) and \(\beta = \frac{\pi}{12}\).
Use the formula to rewrite the expression as \(\cos\left(\frac{5\pi}{12} - \frac{\pi}{12}\right)\).
Simplify the angle inside the cosine: \(\frac{5\pi}{12} - \frac{\pi}{12} = \frac{4\pi}{12} = \frac{\pi}{3}\).
Recognize that the expression equals \(\cos\left(\frac{\pi}{3}\right)\), and recall the exact value of \(\cos\left(\frac{\pi}{3}\right)\) from the unit circle.

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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 between two angles α and β is given by cos(α - β) = cos α cos β + sin α sin β. This identity allows us to rewrite expressions involving sums of products of sines and cosines as a single cosine function, simplifying evaluation.
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Verifying Identities with Sum and Difference Formulas

Exact Values of Trigonometric Functions at Special Angles

Certain angles, especially multiples of π/6, π/4, and π/3, have well-known exact sine and cosine values. Recognizing these angles helps in calculating exact trigonometric values without a calculator, which is essential for precise answers.
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Introduction to Trigonometric Functions

Angle Simplification and Periodicity

Trigonometric functions are periodic, so angles can be simplified by adding or subtracting multiples of 2π to find equivalent angles within a standard interval. This simplification aids in evaluating trigonometric expressions accurately.
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Period of Sine and Cosine Functions
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

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. Write the expression as the cosine of an angle.

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

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

Use the given information to find the exact value of each of the following: cos2θ\(\cos\)2\(\theta\)

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

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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