Question

find ( mangle x ).
write your answer as an integer or as a decimal rounded to the nearest tenth.
( mangle x=) 。

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \(a^{2}=b^{2}+c^{2}-2bc\cos A\). In \(\triangle WXY\), let \(WY = 8\), \(WX=13\), \(YX = 19\). We want to find \(\angle X\), so we use the formula \(WY^{2}=WX^{2}+YX^{2}-2\cdot WX\cdot YX\cdot\cos X\).
Substitute the values: \(8^{2}=13^{2}+19^{2}-2\times13\times19\times\cos X\).

$$64 = 169+361-494\cos X$$

$$64=530 - 494\cos X$$

Step2: Solve for \(\cos X\)

Rearrange the equation:

$$494\cos X=530 - 64$$

$$494\cos X = 466$$

\(\cos X=\frac{466}{494}\approx0.9433\)

Step3: Find \(X\)

Take the inverse - cosine: \(X=\cos^{-1}(0.9433)\)
Using a calculator, \(X\approx19.4^{\circ}\)

Answer:

\(19.4\)