Question

find ( mangle d ).

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

Explanation:

Step1: Identify triangle type and trigonometric ratio

Triangle \( CDE \) is a right triangle with \( \angle C = 90^\circ \), \( CD = 3 \), and \( DE = 4 \). To find \( m\angle D \), we can use the cosine ratio: \( \cos(\angle D)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{CD}{DE} \).

Step2: Substitute values and calculate

Substitute \( CD = 3 \) and \( DE = 4 \) into the cosine formula: \( \cos(\angle D)=\frac{3}{4} = 0.75 \). Then, find the angle whose cosine is \( 0.75 \) using the inverse cosine function: \( \angle D=\cos^{-1}(0.75) \).

Step3: Compute the angle

Using a calculator, \( \cos^{-1}(0.75)\approx41.4^\circ \) (rounded to the nearest tenth).

Answer:

\( 41.4 \)