Chapter 10.1, Problem 30E

To determine

To find the sine and cosine ratios for both the acute angles

Answer to Problem 30E

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.

Explanation of Solution

Given information:

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

Calculation:

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

As known, $\sin e=\frac{perpendicular}{hypotenuse}$ , $\cos ine=\frac{base}{hypotenuse}$

The hypotenuse is equal to DE = 36

For angle D, perpendicular = EF = 36 and base = DF = 15

Hence,

  $\mathrm{Sin}D=\frac{EF}{DE}=\frac{36}{39}=\frac{12}{13}$ and $\mathrm{Cos}D=\frac{DF}{DE}=\frac{15}{39}=\frac{5}{13}$

For angle E,

Perpendicular = DF = 15 and base = EF = 36

Hence,

  $\mathrm{Sin}E=\frac{DF}{DE}=\frac{15}{39}=\frac{5}{13}$ and $\mathrm{Cos}E=\frac{EF}{DE}=\frac{36}{39}=\frac{12}{13}$

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