Question
The hypotenuse is labeled as c, which is the longest side of the triangle. The two shorter sides are labeled a (opposite side to the angle θ) and b (adjacent side to θ). The right angle is clearly marked, where the sides a and b meet. There's an angle labeled θ opposite the side a, and another angle ϕ
opposite the side b.
If c = 17 in. and b = 15 in., find the value of sin0. Write your answer as a fraction. You do not need to simplify.
opposite the side b.
If c = 17 in. and b = 15 in., find the value of sin0. Write your answer as a fraction. You do not need to simplify.
Answers
GPT-4o mini
To find the value of \( \sin(θ) \), we can use the definition of sine in a right triangle:
\[
\sin(θ) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
In this case, the side opposite \( θ \) is \( a \), and the hypotenuse is \( c \). We know that \( c = 17 \) inches. However, we need to find the length of side \( a \) first.
We can use the Pythagorean theorem to find side \( a \). According to the Pythagorean theorem:
\[
c^2 = a^2 + b^2
\]
Given \( c = 17 \) inches and \( b = 15 \) inches, we can substitute these values into the equation:
\[
17^2 = a^2 + 15^2
\]
Calculating the squares:
\[
289 = a^2 + 225
\]
Now, we can solve for \( a^2 \):
\[
a^2 = 289 - 225
\]
\[
a^2 = 64
\]
Taking the square root of both sides gives:
\[
a = 8 \text{ in.}
\]
Now that we have the length of side \( a \), we can find \( \sin(θ) \):
\[
\sin(θ) = \frac{a}{c} = \frac{8}{17}
\]
Thus, the value of \( \sin(θ) \) is:
\[
\boxed{\frac{8}{17}}
\]
\[
\sin(θ) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
In this case, the side opposite \( θ \) is \( a \), and the hypotenuse is \( c \). We know that \( c = 17 \) inches. However, we need to find the length of side \( a \) first.
We can use the Pythagorean theorem to find side \( a \). According to the Pythagorean theorem:
\[
c^2 = a^2 + b^2
\]
Given \( c = 17 \) inches and \( b = 15 \) inches, we can substitute these values into the equation:
\[
17^2 = a^2 + 15^2
\]
Calculating the squares:
\[
289 = a^2 + 225
\]
Now, we can solve for \( a^2 \):
\[
a^2 = 289 - 225
\]
\[
a^2 = 64
\]
Taking the square root of both sides gives:
\[
a = 8 \text{ in.}
\]
Now that we have the length of side \( a \), we can find \( \sin(θ) \):
\[
\sin(θ) = \frac{a}{c} = \frac{8}{17}
\]
Thus, the value of \( \sin(θ) \) is:
\[
\boxed{\frac{8}{17}}
\]