Chapter 13.7, Problem 31WE

(a)

To determine

To find: The value of given function.

Answer to Problem 31WE

The value of the given function is $\frac{36}{85}$ .

Explanation of Solution

Given:

The value $\sin \alpha =\frac{-3}{5}$ and $\cos \beta =\frac{8}{17}$ .

The given function is,

  $\sin \left(\alpha +\beta \right)$

The angle $\alpha$ is in the third quadrant and $\beta$ is in the fourth quadrant.

Calculation:

Consider the given function is,

  $\sin \left(\alpha +\beta \right)$

Consider the value of $\cos \alpha$ is determined as,

  $\begin{array}{c}\cos \alpha =\sqrt{1-{\sin }^2\alpha }\\ =\sqrt{1-{\left(\frac{-3}{5}\right)}^2}\\ =\sqrt{\frac{16}{25}}\\ =\frac{4}{5}\end{array}$

The angle $\alpha$ is in the third quadrant.

  $\cos \alpha =-\frac{4}{5}$

Consider the value of $\sin \beta$ is determined as,

  $\begin{array}{c}\sin \beta =\sqrt{1-{\cos }^2\alpha }\\ =\sqrt{1-{\left(\frac{8}{17}\right)}^2}\\ =\sqrt{\frac{225}{289}}\\ =\frac{15}{17}\end{array}$

The angle $\beta$ is in the fourth quadrant.

  $\sin \beta =-\frac{15}{17}$

Expand the above function as,

  $\begin{array}{c}\sin \left(\alpha +\beta \right)=\sin \alpha \cos \beta +\sin \beta \cos \alpha \\ =\left(\frac{-3}{5}\right)\left(\frac{8}{17}\right)+\left(-\frac{4}{5}\right)\left(-\frac{15}{17}\right)\\ =\frac{-24}{85}+\frac{60}{85}\\ =\frac{36}{85}\end{array}$

(b)

To determine

To find: The value of given function.

Answer to Problem 31WE

The value of the given function is $\frac{-77}{85}$ .

Explanation of Solution

Given:

The value $\sin \alpha =\frac{-3}{5}$ and $\cos \beta =\frac{8}{17}$ .

The given function is,

  $\cos \left(\alpha +\beta \right)$

The angle $\alpha$ is in the third quadrant and $\beta$ is in the fourth quadrant.

Calculation:

Consider the given function is,

  $\cos \left(\alpha +\beta \right)$

Then,

  $\begin{array}{c}\cos \left(\alpha +\beta \right)=\cos \alpha \cos \beta -\sin \alpha \sin \beta \\ =\left(\frac{-4}{5}\right)\left(\frac{8}{17}\right)-\left(\frac{-3}{5}\right)\left(-\frac{15}{17}\right)\\ =\frac{-77}{85}\end{array}$

(c)

To determine

To find: The find the quadrant of $\alpha +\beta$ .

Answer to Problem 31WE

The angle $\alpha +\beta$ is the second quadrant.

Explanation of Solution

Consider the given equation is,

  $\begin{array}{l}\sin \alpha =\left(\frac{3}{5}\right)\\ \alpha ={\sin }^{-1}\left(\frac{3}{5}\right)\\ \alpha =36.87°\end{array}$

Consider the given equation is,

  $\begin{array}{l}\cos \beta =\left(\frac{8}{17}\right)\\ \beta ={\cos }^{-1}\left(\frac{8}{17}\right)\\ \beta =98.80°\end{array}$

Then, the value of $\alpha +\beta$ is,

  $\begin{array}{c}\alpha +\beta =98.80°+36.87°\\ =135.67°\end{array}$

Thus, the angle $\alpha +\beta$ is the second quadrant.