Chapter 4.4, Problem 120E

(a)

To determine

To calculate:The exact value of function $\left(f+g\right)\left(\theta \right)$ for $\theta =180°$ , where $f\left(\theta \right)=\sin \theta$ and $g\left(\theta \right)=\cos \theta$ .

Answer to Problem 120E

The value $\left(f+g\right)\left(180°\right)$ is $-1$ .

Explanation of Solution

Given:

The functions $f\left(\theta \right)=\sin \theta$ and $g\left(\theta \right)=\cos \theta$ .

Formula Used:

Use the property $\left(f+g\right)\left(\theta \right)=f\left(\theta \right)+g\left(\theta \right)$ , for the given value of $\theta$ .

Calculation:

Substitute the values of $\sin \theta$ for $f\left(\theta \right)$ and $\cos \theta$ for $g\left(\theta \right)$ in above identity.

  $\begin{array}{c}\left(f+g\right)\left(\theta \right)=f\left(\theta \right)+g\left(\theta \right)\\ =\sin \theta +\cos \theta \end{array}$

Substitute $180°$ for $\theta$ in the above equation.

  $\left(f+g\right)\left(180°\right)=\sin 180°+\cos 180°$

Substitute the trigonometric value $0$ for $\sin \left(180°\right)$ and $-1$ for $\cos \left(180°\right)$ .

  $\begin{array}{c}\left(f+g\right)\left(180°\right)=\sin 180°+\cos 180°\\ =0+\left(-1\right)\\ =-1\end{array}$

Therefore the value $\left(f+g\right)\left(180°\right)$ is $-1$ .

(b)

To determine

To calculate:The exact value of function $\left(g-f\right)\left(\theta \right)$ for $\theta =180°$ , where $f\left(\theta \right)=\sin \theta$ and $g\left(\theta \right)=\cos \theta$ .

Answer to Problem 120E

The value $\left(g-f\right)\left(180°\right)$ is $-1$ .

Explanation of Solution

Given:

The functions $f\left(\theta \right)=\sin \theta$ and $g\left(\theta \right)=\cos \theta$ .

Formula Used:

Use the property $\left(g-f\right)\left(\theta \right)=g\left(\theta \right)-f\left(\theta \right)$ , for the given value of $\theta$ .

Calculation:

Substitute the values of $\sin \theta$ for $f\left(\theta \right)$ and $\cos \theta$ for $g\left(\theta \right)$ in above identity.

  $\begin{array}{c}\left(g-f\right)\left(\theta \right)=g\left(\theta \right)-f\left(\theta \right)\\ =\cos \theta -\sin \theta \end{array}$

Substitute $180°$ for $\theta$ in the above equation.

  $\left(g-f\right)\left(180°\right)=\cos 180°-\sin 180°$

Substitute the trigonometric value $0$ for $\sin \left(180°\right)$ and $-1$ for $\cos \left(180°\right)$ .

  $\begin{array}{c}\left(g-f\right)\left(180°\right)=\cos 180°-\sin 180°\\ =-1-0\\ =-1\end{array}$

Therefore the value $\left(g-f\right)\left(180°\right)$ is $-1$ .

(c)

To determine

To calculate:The exact value of function ${\left[g\left(\theta \right)\right]}^2$ for $\theta =180°$ , where $g\left(\theta \right)=\cos \theta$ .

Answer to Problem 120E

The value ${\left[g\left(180°\right)\right]}^2$ is $1$ .

Explanation of Solution

Given:

The functions $f\left(\theta \right)=\sin \theta$ and $g\left(\theta \right)=\cos \theta$ .

Formula Used:

Use the value of $g\left(\theta \right)=\cos \theta$ in the required value ${\left[g\left(\theta \right)\right]}^2$ , for the given value of $\theta$ .

Calculation:

Substitute the values of $\cos \theta$ for $g\left(\theta \right)$ in above identity.

  ${\left[g\left(\theta \right)\right]}^2={\left(\cos \theta \right)}^2$

Substitute $180°$ for $\theta$ in the above equation.

  ${\left[g\left(180°\right)\right]}^2={\left(\cos 180°\right)}^2$

Now substitute trigonometric value $-1$ for $\cos 180°$ in the above equation.

  $\begin{array}{c}{\left[g\left(180°\right)\right]}^2={\left(\cos 180°\right)}^2\\ ={\left(-1\right)}^2\\ =1\end{array}$

Therefore the value ${\left[g\left(180°\right)\right]}^2$ is $1$ .

(d)

To determine

To calculate:The exact value of function $\left(fg\right)\left(\theta \right)$ for $\theta =180°$ , where $f\left(\theta \right)=\sin \theta$ and $g\left(\theta \right)=\cos \theta$ .

Answer to Problem 120E

The value $\left(fg\right)\left(180°\right)$ is $0$ .

Explanation of Solution

Given:

The functions $f\left(\theta \right)=\sin \theta$ and $g\left(\theta \right)=\cos \theta$ .

Formula Used:

Use the property $\left(fg\right)\left(\theta \right)=f\left(\theta \right)g\left(\theta \right)$ , for the given value of $\theta$ .

Calculation:

Substitute the values $\sin \theta$ for $f\left(\theta \right)$ and $\cos \theta$ for $g\left(\theta \right)$ in above identity.

  $\begin{array}{c}\left(fg\right)\left(\theta \right)=f\left(\theta \right)g\left(\theta \right)\\ =\sin \theta \cos \theta \end{array}$

Substitute $180°$ for $\theta$ in the above equation.

  $\left(fg\right)\left(180°\right)=\sin \left(180°\right)\cos \left(180°\right)$

Now substitute $0$ for $\sin 180°$ and $-1$ for $\cos 180°$ in the above equation.

  $\begin{array}{c}\left(fg\right)\left(180°\right)=\sin \left(180°\right)\cos \left(180°\right)\\ =-1\times 0\\ =0\end{array}$

Therefore the value $\left(fg\right)\left(180°\right)$ for $f\left(\theta \right)=\sin \theta$ and $g\left(\theta \right)=\cos \theta$ is $0$ .

(e)

To determine

To calculate:The exact value of function $f\left(2\theta \right)$ for $\theta =180°$ , where $f\left(\theta \right)=\sin \theta$ .

Answer to Problem 120E

The value $f\left(2\theta \right)$ for $\theta =180°$ is $0$ .

Explanation of Solution

Given:

The functions $f\left(\theta \right)=\sin \theta$ and $g\left(\theta \right)=\cos \theta$ .

Formula Used:

Firstly calculate the angle $2\theta$ for given $\theta$ and then calculate $f\left(2\theta \right)$ .

Calculation:

As $\theta =180°$ , so $2\theta =360°$ .

Substitute the value $360°$ for $2\theta$ in $f\left(2\theta \right)=\sin \left(2\theta \right)$

  $f\left(360°\right)=\sin 360°$

Now substitute trigonometric value $0$ for $\sin 360°$ .

  $\sin \left(360°\right)=0$

Therefore value $f\left(2\theta \right)$ for $\theta =180°$ is $0$ .

(f)

To determine

To calculate:The exact value of function $g\left(-\theta \right)$ for $\theta =180°$ , where $g\left(\theta \right)=\cos \theta$ .

Answer to Problem 120E

The value $g\left(-180°\right)$ is $-1$ .

Explanation of Solution

Given:

The functions $f\left(\theta \right)=\sin \theta$ and $g\left(\theta \right)=\cos \theta$ .

Fomula Used:

Use the property $g\left(-\theta \right)=g\left(\theta \right)$ for the given even function $g\left(\theta \right)=\cos \theta$ , for the given value of $\theta$ .

Calculation:

As by the property of even function $\cos \left(-\theta \right)=\cos \theta$ , now substitute $180°$ for $\theta$ in the equation

  $\cos \left(-180°\right)=\cos \left(180°\right)$

Substitute the trigonometric value $-1$ for $\cos \left(180°\right)$ in the above equation.

  $\cos \left(-180°\right)=-1$

Therefore the value of $\cos \left(-\theta \right)$ for $\theta =180°$ is $-1$ .