Question

find ( mangle c ).
round to the nearest degree.
(image of triangle with vertices b, c, a; side bc = 5, ac = 12, ab = 15)

Explanation:

Step1: Identify the Law to Use

We have a triangle with sides \( a = 15 \), \( b = 12 \), \( c = 5 \), and we need to find the measure of angle \( C \) (opposite side \( c = 5 \)). We use the Law of Cosines, which is \( c^{2}=a^{2}+b^{2}-2ab\cos(C) \).

Step2: Rearrange the Formula for \( \cos(C) \)

From \( c^{2}=a^{2}+b^{2}-2ab\cos(C) \), we can rearrange to solve for \( \cos(C) \):
\[
\cos(C)=\frac{a^{2}+b^{2}-c^{2}}{2ab}
\]
Substitute \( a = 15 \), \( b = 12 \), \( c = 5 \) into the formula:
\[
\cos(C)=\frac{15^{2}+12^{2}-5^{2}}{2\times15\times12}
\]

Step3: Calculate the Numerator and Denominator

First, calculate the numerator: \( 15^{2}=225 \), \( 12^{2}=144 \), \( 5^{2}=25 \). So, \( 225 + 144-25=225 + 119 = 344 \).
Then, calculate the denominator: \( 2\times15\times12 = 360 \).

Step4: Find \( \cos(C) \) and Then \( C \)

\[
\cos(C)=\frac{344}{360}\approx0.9556
\]
Now, find \( C \) by taking the inverse cosine: \( C=\cos^{-1}(0.9556) \). Using a calculator, \( C\approx17^{\circ} \) (rounded to the nearest degree).

Answer:

\( 17^{\circ} \)