Ch. 3 - Trigonometric Identities and Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 3 - Trigonometric Identities and Equations

Problem 40

Chapter 3, Problem 40

In Exercises 39–46, use a half-angle formula to find the exact value of each expression. cos 22.5°

Verified step by step guidance

Recognize that 22.5° is half of 45°, so you can use the half-angle formula for cosine: $\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos\theta}{2}}$.

Identify $\theta = 45^\circ$, so the expression becomes $\cos 22.5^\circ = \pm \sqrt{\frac{1 + \cos 45^\circ}{2}}$.

Recall the exact value of $\cos 45^\circ$, which is $\frac{\sqrt{2}}{2}$.

Substitute $\cos 45^\circ = \frac{\sqrt{2}}{2}$ into the formula: $\cos 22.5^\circ = \pm \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}$.

Determine the correct sign for the square root based on the quadrant where 22.5° lies (first quadrant, where cosine is positive), so choose the positive root.

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Key Concepts

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

Half-Angle Formulas

Half-angle formulas express trigonometric functions of half an angle in terms of the function of the original angle. For cosine, the formula is cos(θ/2) = ±√[(1 + cos θ)/2]. The sign depends on the quadrant of the half-angle. These formulas help find exact values for angles like 22.5°, which is half of 45°.

Exact Values of Common Angles

Knowing the exact trigonometric values of standard angles such as 0°, 30°, 45°, 60°, and 90° is essential. For example, cos 45° = √2/2. These values serve as the basis for applying half-angle formulas to find exact values of non-standard angles like 22.5°.

Sign Determination in Trigonometric Functions

When using half-angle formulas, determining the correct sign (positive or negative) is crucial. This depends on the quadrant where the half-angle lies. Since 22.5° is in the first quadrant, where cosine is positive, the positive root is chosen for cos 22.5°.

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Related Practice

Textbook Question

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 25–32, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. sin 40° cos 20° + cos 40° sin 20°

Textbook Question

In Exercises 59–68, verify each identity. $\cot \frac{x}{2} = \frac{\sin x}{1 - \cos x}$

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Textbook Question

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Textbook Question

Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). sin 2x = √3 / 2

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Textbook Question

In Exercises 57–64, find the exact value of the following under the given conditions: b. sin (α + β), sin α = 5/6 , 𝝅/2 < α < 𝝅 , and tan β = 3/7 , 𝝅 < β < 3𝝅/2 .

Textbook Question

Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). cos 4x = ﹣√3 / 2

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