Question

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

Explanation:

Step1: Identify the trigonometric ratio

In right - triangle \(ABC\) with \(\angle B = 90^{\circ}\), we know the adjacent side \(BC = 6\) to \(\angle C\) and the hypotenuse \(AC=8\). We use the cosine ratio: \(\cos C=\frac{\text{adjacent}}{\text{hypotenuse}}\).
So, \(\cos C=\frac{BC}{AC}\).

Step2: Substitute the values

Substitute \(BC = 6\) and \(AC = 8\) into the cosine formula: \(\cos C=\frac{6}{8}=0.75\).

Step3: Find the angle

To find \(m\angle C\), we take the inverse cosine (arccos) of \(0.75\). So, \(m\angle C=\cos^{- 1}(0.75)\).
Using a calculator, \(\cos^{-1}(0.75)\approx41.4^{\circ}\).

Answer:

\(41.4\)