Question

This image shows a right triangle with sides labeled as \a\, \b\, and \c\. The right angle is at the vertex between sides \a\ and \b\. The angle opposite side \a\ is labeled as \\\theta\, and the angle opposite side \b\ is labeled as \\\phi\. This image is relevant for studying trigonometry and geometry, as it illustrates the relationships between the sides and angles of a right triangle.

If c=17 in. and b=15 in., find the value of sinθ. Write your answer as a fraction. You do not need to simplify.

(1 point)

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Answer 1

To find \( \sin \theta \) in the right triangle with the sides labeled, we can use the definition of sine in terms of a right triangle.

The sine of an angle \( \theta \) is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In this case:

- The side opposite angle \( \theta \) is \( a \),
- The hypotenuse is \( c \) (which is given as 17 in).

We first need to find the length of side \( a \) using the Pythagorean theorem, which states:

\[
c^2 = a^2 + b^2
\]

Substituting the known values:

\[
17^2 = a^2 + 15^2
\]

Calculating \( 17^2 \) and \( 15^2 \):

\[
289 = a^2 + 225
\]

Now, solve for \( a^2 \):

\[
a^2 = 289 - 225 = 64
\]

Taking the square root to find \( a \):

\[
a = \sqrt{64} = 8 \text{ in.}
\]

Now we can find \( \sin \theta \):

\[
\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{a}{c} = \frac{8}{17}
\]

Thus, the value of \( \sin \theta \) is

\[
\frac{8}{17}.
\]