Question

Triangle:
5
4
3
θ

Use the inverse if cosine to solve for the missing angle, rounded to the nearest whole degree.

a. 90 degree
b. 53 degree
c. 37 degree
d. 1 degree

Answer 1

To find the missing angle \( \theta \) in the triangle with sides 5, 4, and 3, we can use the cosine rule:

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

where \( a \), \( b \), and \( c \) are the lengths of the sides of the triangle opposite to angles \( A \), \( B \), and \( C \) respectively. Let's assign:

- \( a = 5 \)
- \( b = 4 \)
- \( c = 3 \)

Using the values, we plug in the numbers into the formula:

\[
\cos(\theta) = \frac{5^2 + 4^2 - 3^2}{2 \cdot 5 \cdot 4}
\]

Calculating the values:

\[
\cos(\theta) = \frac{25 + 16 - 9}{40}
\]
\[
\cos(\theta) = \frac{32}{40}
\]
\[
\cos(\theta) = 0.8
\]

Now, we find \( \theta \) using the inverse cosine function:

\[
\theta = \cos^{-1}(0.8)
\]

Using a calculator, we find:

\[
\theta \approx 36.87 \text{ degrees}
\]

Rounding this to the nearest whole degree gives us:

\[
\theta \approx 37 \text{ degrees}
\]

Therefore, the answer is:

**c. 37 degrees**.