V17 Find all possible values of cos(0). 9. Let sin(0)

Algebra and Trigonometry (6th Edition)
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Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Problem Statement:**

Let \(\sin(\theta) = \frac{\sqrt{17}}{9}\). Find all possible values of \(\cos(\theta)\).

**Solution:**

To determine the possible values of \(\cos(\theta)\), we can use the fundamental Pythagorean identity which is:

\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]

Given:
\[
\sin(\theta) = \frac{\sqrt{17}}{9}
\]

First, square both sides to find \(\sin^2(\theta)\):

\[
\sin^2(\theta) = \left(\frac{\sqrt{17}}{9}\right)^2 = \frac{17}{81}
\]

Substitute this back into the Pythagorean identity:

\[
\frac{17}{81} + \cos^2(\theta) = 1
\]

Solve for \(\cos^2(\theta)\):

\[
\cos^2(\theta) = 1 - \frac{17}{81} = \frac{81}{81} - \frac{17}{81} = \frac{64}{81}
\]

Therefore, \(\cos(\theta)\) can be either the positive or negative square root of \(\frac{64}{81}\):

\[
\cos(\theta) = \pm \frac{8}{9}
\]

**Answer:**

\[
\cos(\theta) = \frac{8}{9} \quad \text{or} \quad \cos(\theta) = -\frac{8}{9}
\]

These are the possible values of \(\cos(\theta)\) given that \(\sin(\theta) = \frac{\sqrt{17}}{9}\).
Transcribed Image Text:**Problem Statement:** Let \(\sin(\theta) = \frac{\sqrt{17}}{9}\). Find all possible values of \(\cos(\theta)\). **Solution:** To determine the possible values of \(\cos(\theta)\), we can use the fundamental Pythagorean identity which is: \[ \sin^2(\theta) + \cos^2(\theta) = 1 \] Given: \[ \sin(\theta) = \frac{\sqrt{17}}{9} \] First, square both sides to find \(\sin^2(\theta)\): \[ \sin^2(\theta) = \left(\frac{\sqrt{17}}{9}\right)^2 = \frac{17}{81} \] Substitute this back into the Pythagorean identity: \[ \frac{17}{81} + \cos^2(\theta) = 1 \] Solve for \(\cos^2(\theta)\): \[ \cos^2(\theta) = 1 - \frac{17}{81} = \frac{81}{81} - \frac{17}{81} = \frac{64}{81} \] Therefore, \(\cos(\theta)\) can be either the positive or negative square root of \(\frac{64}{81}\): \[ \cos(\theta) = \pm \frac{8}{9} \] **Answer:** \[ \cos(\theta) = \frac{8}{9} \quad \text{or} \quad \cos(\theta) = -\frac{8}{9} \] These are the possible values of \(\cos(\theta)\) given that \(\sin(\theta) = \frac{\sqrt{17}}{9}\).
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