If sec $\theta$ = $\frac{17}{12}$ , then what is the value of sin $\theta$ ? ($0^o$ < $\theta$ < $90^o$ )
$\frac{\sqrt{145}}{17}$
To find the value of \sin \theta given \sec \theta = \frac{17}{12}, we can use trigonometric identities.
The identity for secant and cosine is:
\sec \theta = \frac{1}{\cos \theta}
Given that \sec \theta = \frac{17}{12}, we can find \cos \theta as follows:
\cos \theta = \frac{1}{\sec \theta} = \frac{1}{\frac{17}{12}} = \frac{12}{17}
Next, use the Pythagorean identity:
\sin^2 \theta + \cos^2 \theta = 1
Substitute the value of \cos \theta:
\sin^2 \theta + \left(\frac{12}{17}\right)^2 = 1
Calculate the square of \cos \theta:
\left(\frac{12}{17}\right)^2 = \frac{144}{289}
Substitute back into the equation:
\sin^2 \theta = 1 - \frac{144}{289}
Simplify:
\sin^2 \theta = \frac{289}{289} - \frac{144}{289} = \frac{145}{289}
To find \sin \theta, take the square root:
\sin \theta = \sqrt{\frac{145}{289}} = \frac{\sqrt{145}}{17}
Therefore, the value of \sin \theta is \frac{\sqrt{145}}{17}, which corresponds to option (c).
Other options are incorrect because:
- \frac{13}{17} does not satisfy the Pythagorean identity when calculated with the given \sec \theta.
- \frac{12}{17} is the value of \cos \theta, not \sin \theta.
- \frac{\sqrt{170}}{17} is not derived from the given conditions.
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