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Chapter 5: Problem 105

In Exercises \(105-108,\) use the trigonometric substitution to write the algebraic equation as a trigonometric equation of \(\theta\) where \(-\pi / 2<\theta<\pi / 2 .\) Then find \(\sin \theta\) and \(\cos \theta\). $$ 3=\sqrt{9-x^{2}}, \quad x=3 \sin \theta $$

Short Answer

Expert verified
\(\sin \theta = 1\) and \(\cos \theta = 0\)

Step by step solution

01

Substitution

Substitute \( x = 3 \sin \theta \) into the equation to get \( 3 = \sqrt{9 - (3 \sin \theta)^2} \). After this, simplify the equation.
02

Simplify the Equation

Applying the square rule to the right side of the equation, it becomes \( 3 = \sqrt{9 - 9 \sin^2 \theta} \). Then we can rewrite the right side of the equation using the Pythagorean Trigonometric Identity \( \cos^2 \theta = 1 - \sin^2 \theta \) to get \( 3 = \sqrt{9 - 9 \cos^2 \theta} \). Dividing throughout by 3, we obtain \( 1 = \sqrt{1 - \cos^2 \theta} \), since \( 1 = \sin \theta \) in this case, we can now find \( \cos \theta \) by using the Pythagorean Trigonometric Identity \( \sin^2 \theta + \cos^2 \theta = 1 \).
03

Find the Value of Cosine

Since by Pythagorean trigonometric identity, we know that \( \sin^2 \theta + \cos^2 \theta = 1 \), we can rearrange it to \( \cos^2 \theta = 1 - \sin^2 \theta \). Hence, \( \cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 -1} = 0 \).

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Most popular questions from this chapter

In Exercises 125 - 128, use a graphing utility to verify the identity. Confirm that it is an identity algebraically. \( \left(\cos 3x - \cos x\right) / \left(\sin 3x - \sin x\right) = -\tan 2x \)

In Exercises 111 - 124, verify the identity. \( \sec 2 \theta = \dfrac{\sec^2 \theta}{2 - \sec^2 \theta} \)

In Exercises 37-42, find the exact values of \( \sin 2u \), \( \cos 2u \), and \( \tan 2u \) using the double-angle formulas. \( \sec u = - 2, \dfrac{\pi}{2} < u < \pi \)

In Exercises 111 - 124, verify the identity. \( \left(\sin x + \cos x\right)^2 = 1 + \sin 2x \)

Exercises 43-52, use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. \( \sin^2 x \cos^4 x \)
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