Chapter 7, Problem 8CRT

(a)

To determine

To find: The side $x$ labeled in the diagram.

Answer to Problem 8CRT

The exact value of $\cos \frac{7\pi }{6}$ is $-\frac{\sqrt{3}}{2}$ .

Explanation of Solution

Given information:

The diagram of the given triangle is:

  

Calculation:

As the sum of all the interior angles of the triangle is $180°$ . This gives the third angle of the triangle equals to $180°-70°-65°=45°$ .

Now use the formula.

  $\frac{\sin 45°}{5.6}=\frac{\sin 65°}{x}$

Now substitute $\frac{1}{\sqrt{2}}$ for $\sin 45°$ and $0.91$ for $\sin 65°$ in the formula.

  $\begin{array}{c}\frac{\sin 45°}{5.6}=\frac{\sin 65°}{x}\\ \frac{\frac{1}{\sqrt{2}}}{5.6}=\frac{0.91}{x}\\ x=0.91\times 5.6\times \sqrt{2}\\ x=7.21\end{array}$

Therefore, The value of $x$ is $7.21$ approx.

(b)

To determine

To find: The angle $x$ labeled in the diagram.

Answer to Problem 8CRT

The value of the angle $x$ is $1.62$ radians.

Explanation of Solution

Given information:

The diagram of the given triangle is:

  

Calculation:

As per the diagram the sides of the triangle $a=15.2$ , $b=13.0$ and $c=20.5$ .

Use the cosine law for triangle.

  $\begin{array}{c}{c}^2=a^2+b^2-2ab\cos x\\ {20.5}^2={15.2}^2+{13}^2-2\times 15.2\times 13\times \cos x\\ 420.25=231.04+169-395.2\cos x\\ 395.2\cos x=-20.21\end{array}$

Further simplify,

  $\begin{array}{c}395.2\cos x=-20.21\\ \cos x=\frac{-20.21}{395.2}\\ x={\cos }^{-1}\left(\frac{-20.21}{395.2}\right)\\ x=1.62\end{array}$

Therefore, the value of the angle $x$ is $1.62$ radians.