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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 3.3.58b
Chapter 3, Problem 3.3.58b

In Exercises 55–58, use the given information to find the exact value of each of the following:
b. cos(α/2)
sec α = ﹣3, 𝝅/2 < α < 𝝅

Verified step by step guidance
1
Identify the given information: \( \sec \alpha = -3 \) and the interval \( \frac{\pi}{2} < \alpha < \pi \). Recall that \( \sec \alpha = \frac{1}{\cos \alpha} \).
Find \( \cos \alpha \) by taking the reciprocal of \( \sec \alpha \): \( \cos \alpha = \frac{1}{\sec \alpha} = \frac{1}{-3} = -\frac{1}{3} \).
Since \( \alpha \) is in the interval \( \frac{\pi}{2} < \alpha < \pi \), which corresponds to the second quadrant, note that cosine values are negative there, confirming the sign of \( \cos \alpha \).
Use the double-angle formula for cosine to find \( \cos 2\alpha \): \[ \cos 2\alpha = 2 \cos^2 \alpha - 1 \].
Substitute \( \cos \alpha = -\frac{1}{3} \) into the double-angle formula and simplify the expression to find the exact value of \( \cos 2\alpha \).

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Key Concepts

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

Secant and Cosine Relationship

Secant (sec) is the reciprocal of cosine (cos), so sec α = 1/cos α. Given sec α = -3, we find cos α by taking the reciprocal, resulting in cos α = -1/3. Understanding this reciprocal relationship is essential to convert between secant and cosine values.
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Graphs of Secant and Cosecant Functions

Angle Interval and Sign of Trigonometric Functions

The interval π/2 < α < π places α in the second quadrant, where cosine values are negative. This information confirms the sign of cos α, ensuring the correct value is chosen when solving for cosine or related functions.
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Introduction to Trigonometric Functions

Half-Angle Formulas

To find cos(α/2), use the half-angle identity: cos(α/2) = ±√[(1 + cos α)/2]. The sign depends on the quadrant of α/2. Applying this formula allows calculation of the exact value of cos(α/2) from the known cos α.
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Quadratic Formula
Related Practice
Textbook Question

In Exercises 57–64, find the exact value of the following under the given conditions:

c. tan (α + β)

cos α = 8/17, α lies in quadrant IV, and sin β = -1/2, β lies in quadrant III.

5
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Textbook Question

In Exercises 57–64, find the exact value of the following under the given conditions:

c. tan (α + β)

sin α = 3/5, α lies in quadrant I, and sin β = 5/13, β lies in quadrant II.

1
views
Textbook Question

In Exercises 57–64, find the exact value of the following under the given conditions:

b. sin (α + β)

sin α = 3/5, α lies in quadrant I, and sin β = 5/13, β lies in quadrant II.

Textbook Question

In Exercises 55–58, use the given information to find the exact value of each of the following:

c. tan(α/2)

tan α = 4/3, 180° < α < 270°

5
views
Textbook Question

In Exercises 55–58, use the given information to find the exact value of each of the following:

b. cos(α/2)

tan α = 4/3, 180° < α < 270°

Textbook Question

In Exercises 57–64, find the exact value of the following under the given conditions:

b. sin (α + β)

cos α = 8/17, α lies in quadrant IV, and sin β = -1/2, β lies in quadrant III.