Chapter 12, Problem 1PT

To determine

To calculate: Unknown sides and angle of the given triangle.

Answer to Problem 1PT

The side lengths $a=5.2$ and $b=7.2$ and the angle $B=54°$

Explanation of Solution

Given information: A diagram of right-angled triangle is given, that means $C=90°$ . Also, $A=36°$ and $c=9$given.

  

Formula used: We know the sum of the interior angle of the triangle $A+B+C=180°$ . Also, for angle $A$ in the given triangle, $\sin (A)=\left(\frac{a}{c}\right)$ . And $\cos (A)=\left(\frac{b}{c}\right)$ . Also,

Calculation: Find the angle $B$ ,

  $\begin{array}{l}A+B+C=180°\\ 36+B+90=180\\ B+126=180\\ B=180-126\\ B=54°\end{array}$

Now,

Using $\sin (A)=\left(\frac{a}{c}\right)$ , find $a$ .

  $\begin{array}{l}\sin (A)=\left(\frac{a}{c}\right)\\ \sin (36)=\left(\frac{a}{9}\right)\\ 0.58=\frac{a}{9}\\ a=9\times 0.58\\ a=5.22\\ a=5.2\end{array}$

Now, using $\cos (A)=\left(\frac{b}{c}\right)$ , find $b$ .

  $\begin{array}{l}\cos (A)=\left(\frac{b}{c}\right)\\ \cos (36)=\left(\frac{b}{9}\right)\\ 0.8=\frac{b}{9}\\ b=9\times 0.8\\ b=7.2\end{array}$