Question

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

Explanation:

Step1: Use the Law of Cosines

The Law of Cosines formula is \( y^{2}=x^{2}+z^{2}-2xz\cos Y\). For \(\triangle XYZ\), to find \(\angle X\), we use \( YZ^{2}=XZ^{2}+XY^{2}-2(XZ)(XY)\cos X\). Let \(XZ = 6\), \(XY=12\), \(YZ = 11\). Then \(11^{2}=6^{2}+12^{2}-2\times6\times12\times\cos X\).

Step2: Simplify the equation

First, calculate the squares: \(121=36 + 144-144\cos X\). Then \(121=180-144\cos X\).

Step3: Solve for \(\cos X\)

Rearrange the equation: \(144\cos X=180 - 121\). So \(144\cos X = 59\), and \(\cos X=\frac{59}{144}\approx0.4097\).

Step4: Find \(X\)

Take the inverse - cosine: \(X=\cos^{-1}(0.4097)\). Using a calculator, \(X\approx65.8^{\circ}\).

Answer:

\(65.8\)