Question

find ( mangle w ).
write your answer as an integer or as a decimal rounded to the nearest tenth.
( mangle w = square^circ )

Explanation:

Step1: Identify the Law to Use

We have a triangle with sides \( WY = 10 \), \( WX = 9 \), and \( YX = 17 \). To find the measure of angle \( W \), we can use the Law of Cosines, which is given by the formula:
\[
c^2 = a^2 + b^2 - 2ab \cos(C)
\]
where \( C \) is the angle opposite side \( c \), and \( a \) and \( b \) are the other two sides. In this case, angle \( W \) is opposite side \( YX = 17 \), so \( c = 17 \), \( a = 10 \), and \( b = 9 \).

Step2: Substitute Values into the Law of Cosines

Substitute \( a = 10 \), \( b = 9 \), and \( c = 17 \) into the formula:
\[
17^2 = 10^2 + 9^2 - 2(10)(9) \cos(\angle W)
\]
Calculate the squares:
\[
289 = 100 + 81 - 180 \cos(\angle W)
\]
Simplify the right side:
\[
289 = 181 - 180 \cos(\angle W)
\]

Step3: Solve for \( \cos(\angle W) \)

Subtract 181 from both sides:
\[
289 - 181 = -180 \cos(\angle W)
\]
\[
108 = -180 \cos(\angle W)
\]
Divide both sides by -180:
\[
\cos(\angle W) = \frac{108}{-180} = -0.6
\]

Step4: Find the Angle

Take the inverse cosine (arccos) of -0.6 to find the measure of angle \( W \):
\[
\angle W = \arccos(-0.6)
\]
Using a calculator, we find that \( \arccos(-0.6) \approx 126.9^\circ \) (rounded to the nearest tenth).

Answer:

\( 126.9 \)