Question

find sin d, sin e, cos d, and cos e. write each answer as a fraction in simplest form and as a decimal rounded to four places. triangle def with right angle at f, df=45, ef=28, de=53 sin d = \\(\square\\) ≈ \\(\square\\) sin e = \\(\square\\) ≈ \\(\square\\) cos d = \\(\square\\) ≈ 8 \\(\square\\) cos e = \\(\square\\) ≈ \\(\square\\)

Explanation:

Step1: Recall SOHCAHTOA

In a right triangle, \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\) and \(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\).

Step2: Find \(\sin D\)

For \(\angle D\), opposite side is \(EF = 28\), hypotenuse is \(DE=53\). So \(\sin D=\frac{28}{53}\approx0.5283\).

Step3: Find \(\sin E\)

For \(\angle E\), opposite side is \(DF = 45\), hypotenuse is \(DE = 53\). So \(\sin E=\frac{45}{53}\approx0.8491\).

Step4: Find \(\cos D\)

For \(\angle D\), adjacent side is \(DF = 45\), hypotenuse is \(DE = 53\). So \(\cos D=\frac{45}{53}\approx0.8491\).

Step5: Find \(\cos E\)

For \(\angle E\), adjacent side is \(EF = 28\), hypotenuse is \(DE = 53\). So \(\cos E=\frac{28}{53}\approx0.5283\).

Answer:

• \(\sin D=\frac{28}{53}\approx0.5283\)
• \(\sin E=\frac{45}{53}\approx0.8491\)
• \(\cos D=\frac{45}{53}\approx0.8491\)
• \(\cos E=\frac{28}{53}\approx0.5283\)