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.
right triangle def with right angle at f, df=12, ef=35, de=37
sin d =
≈
sin e =
≈
cos d =
≈
cos e =
≈

Explanation:

Step1: Recall SOHCAHTOA for sine and cosine

In a right triangle, \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\) and \(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\). The hypotenuse here is \(DE = 37\), \(EF = 35\), \(DF = 12\).

Step2: Calculate \(\sin D\)

For \(\angle D\), opposite side is \(EF = 35\), hypotenuse is \(37\). So \(\sin D=\frac{35}{37}\approx0.9459\).

Step3: Calculate \(\sin E\)

For \(\angle E\), opposite side is \(DF = 12\), hypotenuse is \(37\). So \(\sin E=\frac{12}{37}\approx0.3243\).

Step4: Calculate \(\cos D\)

For \(\angle D\), adjacent side is \(DF = 12\), hypotenuse is \(37\). So \(\cos D=\frac{12}{37}\approx0.3243\).

Step5: Calculate \(\cos E\)

For \(\angle E\), adjacent side is \(EF = 35\), hypotenuse is \(37\). So \(\cos E=\frac{35}{37}\approx0.9459\).

Answer:

\(\sin D=\frac{35}{37}\approx0.9459\)
\(\sin E=\frac{12}{37}\approx0.3243\)
\(\cos D=\frac{12}{37}\approx0.3243\)
\(\cos E=\frac{35}{37}\approx0.9459\)