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 triangle type and sides

Triangle \( XYW \) is right - angled at \( X \). So, it is a right - triangle. We know that in right - triangle \( XYW \), \( XY \) and \( XW \) are the legs, and \( YW \) is the hypotenuse? Wait, no. Wait, the side opposite to \( \angle W \) is \( XY \), the adjacent side to \( \angle W \) is \( XW = 8 \), and the hypotenuse is \( YW=9 \)? Wait, no, in a right - triangle, for angle \( W \), the opposite side is \( XY \), the adjacent side is \( XW \), and the hypotenuse is \( YW \). But we can also use the sine or cosine function. Wait, first, we can find the length of \( XY \) using the Pythagorean theorem. Let \( XY = a \), \( XW = b = 8 \), \( YW = c = 9 \). Then by Pythagoras, \( a^{2}+b^{2}=c^{2}\), so \( a=\sqrt{c^{2}-b^{2}}=\sqrt{9^{2}-8^{2}}=\sqrt{81 - 64}=\sqrt{17}\approx4.123 \).

Step2: Use trigonometric ratio to find \( \angle W \)

We can use the sine function. The sine of an angle in a right - triangle is defined as \( \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} \). For \( \angle W \), the opposite side is \( XY\approx4.123 \) and the hypotenuse is \( YW = 9 \). So \( \sin(\angle W)=\frac{XY}{YW}=\frac{\sqrt{17}}{9}\approx\frac{4.123}{9}\approx0.458 \). Then \( \angle W=\arcsin(0.458) \). Using a calculator, \( \arcsin(0.458)\approx27.3^{\circ} \) (rounded to the nearest tenth).

Alternatively, we can use the cosine function. The adjacent side to \( \angle W \) is \( XW = 8 \), hypotenuse \( YW = 9 \). So \( \cos(\angle W)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{8}{9}\approx0.8889 \). Then \( \angle W=\arccos(0.8889) \). Let's calculate \( \arccos(0.8889) \). Using a calculator, \( \arccos(0.8889)\approx27.3^{\circ} \) (rounded to the nearest tenth).

Answer:

\( 27.3 \)