Question

side angle of the triangle: 13
side angle of the triangle: 12
side angle of the triangle: 5

Use the inverse of cosine to find the indicated missing angle. Round your answer to the nearest tenth.

θ ≈ ___

a. 22.6
b. 22
c. 22.8
d. 21.6

Answer 1

To find the missing angle (θ) in a triangle using the cosine rule, we can use the formula:

\[
\cos(\theta) = \frac{a^2 + b^2 - c^2}{2ab}
\]

Where:
- \( a \) and \( b \) are the lengths of the sides adjacent to angle \( \theta \).
- \( c \) is the length of the side opposite to angle \( \theta \).

In this case, let:
- \( a = 13 \)
- \( b = 12 \)
- \( c = 5 \)

Plugging these values into the cosine formula, we get:

\[
\cos(\theta) = \frac{13^2 + 12^2 - 5^2}{2 \cdot 13 \cdot 12}
\]

Calculating the squares:

\[
\cos(\theta) = \frac{169 + 144 - 25}{2 \cdot 13 \cdot 12}
\]
\[
= \frac{288}{312}
\]
\[
= \frac{24}{26}
\]
\[
= \frac{12}{13}
\]

Now, we find \( \theta \) by taking the inverse cosine:

\[
\theta = \cos^{-1}\left(\frac{12}{13}\right)
\]

Using a calculator to compute this gives:

\[
\theta \approx 22.6^\circ
\]

Rounding to the nearest tenth, the answer is:

**θ ≈ 22.6 (a)**