Question

Use the inverse of cosine to find the approximate measure of the missing angle θ . (1 point) Responses 59.5° 59.5° 43° 43° 30.5° 30.5° 90°

Sides are 56, 65, and 33
θ is the angle between 56 and 65

Answer 1

To find the angle θ between the sides of lengths 56 and 65, you can use the Law of Cosines, which states:

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

where:
- \( c \) is the length of the side opposite the angle θ (33 in this case),
- \( a \) and \( b \) are the lengths of the other two sides (56 and 65).

We want to find θ, so we rearranging the formula as follows:

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

Now, plug in the values:

\[
a = 56, \quad b = 65, \quad c = 33
\]

Calculate \( a^2 \), \( b^2 \), and \( c^2 \):

\[
a^2 = 56^2 = 3136
\]
\[
b^2 = 65^2 = 4225
\]
\[
c^2 = 33^2 = 1089
\]

Now substitute these values into the formula:

\[
\cos(\theta) = \frac{3136 + 4225 - 1089}{2 \cdot 56 \cdot 65}
\]
\[
\cos(\theta) = \frac{3136 + 4225 - 1089}{7280}
\]
\[
\cos(\theta) = \frac{6262}{7280}
\]
\[
\cos(\theta) \approx 0.8617
\]

Now use the inverse cosine function to find θ:

\[
\theta \approx \cos^{-1}(0.8617)
\]

Calculating that gives:

\[
\theta \approx 30.5^\circ
\]

So the approximate measure of the missing angle θ is **30.5°**.