Question

Write the sine and cosine ratios for both acute angles of the triangle.

Short answer

The sine and cosine ratios for the acute angle D, are $\frac{12}{13}$and $\frac{5}{13}$ respectively, and for acute angle E, are$\frac{5}{13}$ and $\frac{12}{13}$ respectively.

Step-by-step solution

Step 1. Given Information.

All the three sides of the triangle are given as 15, 36, and 39. Angle F is given as a right angle.

Step 2. Calculation.

Since F is a right angle, the two acute angles are D and E.

It is known that, $\sin x=\frac{\text{perpendicular}}{\text{hypotenuse}}$and $\cos x=\frac{\text{base}}{\text{hypotenuse}}$

Hypotenuse is$DE=39$.

For Angle D, perpendicular is $EF=36$and base is$DF=15$.

So:

$\begin{array}{l}\mathrm{Sin}D=\frac{EF}{DE}\\ =\frac{36}{39}\\ =\frac{12}{13}\end{array}$

And,

$\begin{array}{l}\mathrm{Cos}D=\frac{DF}{DE}\\ =\frac{15}{39}\\ =\frac{5}{13}\end{array}$

For Angle E, perpendicular is $DF=15$and base is$EF=36$.

$\begin{array}{l}\mathrm{Sin}E=\frac{DF}{DE}\\ =\frac{15}{39}\\ =\frac{5}{13}\end{array}$

And,

$\begin{array}{l}\mathrm{Cos}E=\frac{EF}{DE}\\ =\frac{36}{39}\\ =\frac{12}{13}\end{array}$

Step 3. Conclusion.

The sine and cosine ratios of the acute angle D, are $\frac{12}{13}$ and $\frac{5}{13}$ respectively, and sine and cosine ratios of the acute angle E, are $\frac{5}{13}$ and $\frac{12}{13}$ respectively..