Question
Use the inverse of cosine. The opposite is 15 and the adjacent is 8. The hypotenuse is 17
Answers
Answered by
GPT-4o mini
To find the angle using the inverse of cosine, you need to understand the relationship between the sides of the right triangle and the cosine function.
In a right triangle, the cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.
Using the triangle you provided:
- Adjacent side = 8
- Hypotenuse = 17
We can use the cosine function as follows:
\[
\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{8}{17}
\]
Next, to find the angle \(\theta\), we take the inverse cosine (also known as arccosine):
\[
\theta = \cos^{-1}\left(\frac{8}{17}\right)
\]
Now, using a calculator to compute \( \theta \):
\[
\theta \approx \cos^{-1}(0.4706) \approx 62.73^\circ
\]
Thus, the angle \(\theta\) is approximately \(62.73^\circ\).
If you also wanted to find the other angle in the triangle, you could use the fact that the angles in a right triangle add up to \(90^\circ\):
\[
90^\circ - \theta \approx 90^\circ - 62.73^\circ \approx 27.27^\circ
\]
You also mentioned the opposite side of 15, which is used to confirm the triangle's right-angle properties, but the calculation for cosine specifically used the adjacent side to derive the angle as shown above.
In a right triangle, the cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.
Using the triangle you provided:
- Adjacent side = 8
- Hypotenuse = 17
We can use the cosine function as follows:
\[
\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{8}{17}
\]
Next, to find the angle \(\theta\), we take the inverse cosine (also known as arccosine):
\[
\theta = \cos^{-1}\left(\frac{8}{17}\right)
\]
Now, using a calculator to compute \( \theta \):
\[
\theta \approx \cos^{-1}(0.4706) \approx 62.73^\circ
\]
Thus, the angle \(\theta\) is approximately \(62.73^\circ\).
If you also wanted to find the other angle in the triangle, you could use the fact that the angles in a right triangle add up to \(90^\circ\):
\[
90^\circ - \theta \approx 90^\circ - 62.73^\circ \approx 27.27^\circ
\]
You also mentioned the opposite side of 15, which is used to confirm the triangle's right-angle properties, but the calculation for cosine specifically used the adjacent side to derive the angle as shown above.