Chapter 13.1, Problem 63E

To determine

To calculate: The sides and angle measure of the right angled triangle where $A=27°\text{ and }b=15.2$ .

  

Answer to Problem 63E

The triangle along with sides and acute angles is,

  

Explanation of Solution

Given information:

The right angle triangle where $A=27°\text{ and }b=15.2$ .

  

Formula used:

The trigonometric ratios for a right angle triangle are defined as,

  $\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}},cos\theta =\frac{\text{adjacent}}{\text{hypotenuse}},\tan \theta =\frac{\text{opposite}}{\text{adjacent}},\csc \theta =\frac{\text{hypotenuse}}{\text{opposite}},\sec \theta =\frac{\text{hypotenuse}}{\text{adjacent}}$ and $\cot \theta =\frac{\text{adjacent}}{\text{opposite}}$ .

Calculation:

Consider the right angle triangle $A=27°\text{ and }b=15.2$ .

  

Observe that the length of adjacent side is 15.2 units.

Label the length of opposite side of the triangle as a and hypotenuse as c .

  

Recall that the trigonometric ratios for a right angle triangle are defined as,

  $\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}},cos\theta =\frac{\text{adjacent}}{\text{hypotenuse}},\tan \theta =\frac{\text{opposite}}{\text{adjacent}},\csc \theta =\frac{\text{hypotenuse}}{\text{opposite}},\sec \theta =\frac{\text{hypotenuse}}{\text{adjacent}}$ and $\cot \theta =\frac{\text{adjacent}}{\text{opposite}}$ .

Apply it, to estimate the value of trigonometric ratios,

The value of cosine function is,

  $\begin{array}{c}cos27°=\frac{\text{adjacent}}{\text{hypotenuse}}\\ cos27°=\frac{15.2}{c}\\ c=\frac{15.2}{cos27°}\\ c\approx 17.1\end{array}$

Therefore, the length of hypotenuse is $17.1$ units.

The value of sine function is,

  $\begin{array}{c}\tan 27°=\frac{\text{opposite}}{\text{base}}\\ \tan 27°=\frac{a}{15.2}\\ a=\left(15.2\right)\tan 27°\\ a\approx 7.7\end{array}$

Therefore, the length of opposite side is $7.7$ units.

Observe that the angle A and B are complementary angles so,

  $\begin{array}{c}\angle A+\angle B=90°\\ 27°+\angle B=90°\\ \angle B=63°\end{array}$

Thus, the triangle so formed is provided below,