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°.
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