Question

applying a trigonometric ratio to find a side length
what is the value of d to the nearest hundredth?
d ≈

Explanation:

Step1: Identify the trigonometric ratio

In right - triangle DEF, we know the adjacent side to angle D ($DE = 7.2$) and we want to find the opposite side ($EF=d$) to angle D. We use the tangent ratio. $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, $\theta = 42^{\circ}$, the adjacent side to $\angle D$ is $DE$ and the opposite side is $EF$. So, $\tan(42^{\circ})=\frac{d}{7.2}$.

Step2: Solve for d

Multiply both sides of the equation $\tan(42^{\circ})=\frac{d}{7.2}$ by 7.2. We get $d = 7.2\times\tan(42^{\circ})$. Since $\tan(42^{\circ})\approx0.9004$, then $d=7.2\times0.9004 = 6.48288$.

Step3: Round to the nearest hundredth

Rounding 6.48288 to the nearest hundredth gives $d\approx6.48$.

Answer:

$6.48$