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:

\( 126.9 \)