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Chapter 7: Problem 19

Use an identity to write each expression as a single trigonometric function value or as a single number. $$1-2 \sin ^{2} 15^{\circ}$$

Short Answer

Expert verified
\( \frac{\sqrt{3}}{2} \)

Step by step solution

01

Identify the Trigonometric Identity

Recognize that the expression can use a double-angle identity. The double-angle identity for cosine is: \ \[ \cos(2\theta) = 1 - 2\sin^2(\theta) \] Here, \theta = 15^{\circ}. This identity allows the given expression to be written in terms of \cos(2 \cdot 15^{\circ}).
02

Rewrite the Expression Using the Identity

Use the double-angle identity to rewrite the expression: \[ 1 - 2\sin^2(15^{\circ}) = \cos(2 \cdot 15^{\circ}) \]
03

Simplify the Expression

Simplify the expression inside the cosine function: \[ 2 \cdot 15^{\circ} = 30^{\circ} \] So, \ \[ 1 - 2\sin^2(15^{\circ}) = \cos(30^{\circ}) \]
04

Find the Value of \cos(30^{\circ})

Recall the exact value of \cos(30^{\circ}) which is: \[ \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \] So, the given expression equals \frac{\sqrt{3}}{2}.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Double-Angle Identity
The double-angle identity is a fundamental concept in trigonometry that simplifies expressions involving trigonometric functions. For the cosine function, the double-angle identity is given as: 鈿� Making this connection allows us to recognize that the expression can be simplified using the double-angle identity for cosine.
Cosine Function
The cosine function measures the horizontal coordinate of a point on a unit circle at a given angle from the positive x-axis. This simplifies to 馃挕 Knowing these values helps easily evaluate trigonometric expressions without a calculator.
Simplifying Trigonometric Expressions
Simplifying trigonometric expressions involves using identities and knowing the values of trigonometric functions. In our case, we transformed To simplify: To further simplify: 馃専 This identity allows the transformation of complex expressions into basic trigonometric values, making calculations more manageable.

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Most popular questions from this chapter

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$3 \csc ^{2} \frac{x}{2}=2 \sec x$$

Verify that each equation is an identity. $$\sin 2 x=\frac{2 \tan x}{1+\tan ^{2} x}$$

Use identities to write each expression as a single function of \(x\) or \(\theta\). $$\sin \left(270^{\circ}-\theta\right)$$

Use a graphing calculator to make a conjecture about whether each equation is an identity. $$\sin x=\sqrt{1-\cos ^{2} x}$$

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$\cos \theta-1=\cos 2 \theta$$
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