Question

applying a trigonometric ratio to find a side length
what is the value of d to the nearest hundredth?
d ≈
diagram: right triangle def with right angle at e, de = 7.2, angle at d is 42°, side ef is d

Explanation:

Step1: Identify trigonometric ratio

In right triangle \( DEF \), \( \angle D = 42^\circ \), \( DE = 7.2 \) (adjacent to \( \angle D \)), and \( d = EF \) (opposite to \( \angle D \)). Use tangent: \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \).
\( \tan(42^\circ)=\frac{d}{7.2} \)

Step2: Solve for \( d \)

Multiply both sides by \( 7.2 \): \( d = 7.2\times\tan(42^\circ) \)
Calculate \( \tan(42^\circ)\approx0.9004 \), so \( d\approx7.2\times0.9004 \)
\( d\approx6.483 \)

Step3: Round to nearest hundredth

\( d\approx6.48 \) (since the thousandth digit is 3, which is less than 5, we round down the hundredth digit).

Answer:

\( 6.48 \)