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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
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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
656
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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\(\cos\) \(\frac{5\pi}{12}\) \(\cos\) \(\frac{\pi}{12}\) + \(\sin\) \(\frac{5\pi}{12}\) \(\sin\) \(\frac{\pi}{12}\)

764
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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.\(\sin\) \(\theta\) = \(\frac{12}{13}\), \(\quad\) \(\theta\) \(\text{ lies in quadrant II.}\)

689
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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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