Questions: Use the trigonometric substitution to write the algebraic equation as a trigonometric equation of (theta), where (-pi / 2<theta<pi / 2). (4=sqrt16-x^2, x=4 sin theta ) (4=sqrt16-(4 sin (theta))^2) Find (sin theta) and (cos theta). (Enter your answers as a comma-separated list.) (sin theta=) (cos theta=)

Use the trigonometric substitution to write the algebraic equation as a trigonometric equation of (theta), where (-pi / 2<theta<pi / 2). (4=sqrt16-x^2, x=4 sin theta ) (4=sqrt16-(4 sin (theta))^2) Find (sin theta) and (cos theta). (Enter your answers as a comma-separated list.) (sin theta=) (cos theta=)
Transcript text: Use the trigonometric substitution to write the algebraic equation as a trigonometric equation of $\theta$, where $-\pi / 2<\theta<\pi / 2$. \[ \begin{array}{r} 4=\sqrt{16-x^{2}}, x=4 \sin \theta \\ 4=\sqrt{16-(4 \sin (\theta))^{2}} \end{array} \] Find $\sin \theta$ and $\cos \theta$. (Enter your answers as a comma-separated list.) \[ \begin{array}{l} \sin \theta=\square \\ \cos \theta=\square \end{array} \]
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Solution

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Solution Steps

To solve the given problem, we will use trigonometric substitution. The substitution given is \( x = 4 \sin \theta \). We will substitute this into the equation and simplify to find \(\sin \theta\). Then, using the Pythagorean identity, we will find \(\cos \theta\).

Step 1: Substitute and Simplify

We start with the equation given by the trigonometric substitution \( x = 4 \sin \theta \). Substituting this into the equation \( 4 = \sqrt{16 - x^2} \) gives us: \[ 4 = \sqrt{16 - (4 \sin \theta)^2} \] This simplifies to: \[ 4 = \sqrt{16 - 16 \sin^2 \theta} \]

Step 2: Solve for \(\sin \theta\)

Squaring both sides of the equation results in: \[ 16 = 16 - 16 \sin^2 \theta \] Rearranging this gives: \[ 16 \sin^2 \theta = 0 \] Thus, we find: \[ \sin \theta = 0 \]

Step 3: Calculate \(\cos \theta\)

Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can find \(\cos \theta\): \[ \cos^2 \theta = 1 - \sin^2 \theta = 1 - 0^2 = 1 \] Therefore, we have: \[ \cos \theta = 1 \]

Final Answer

The values are: \[ \sin \theta = 0, \quad \cos \theta = 1 \] Thus, the final answer is: \[ \boxed{0, 1} \]

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