Chapter 1, Problem 7SP

a

To determine

To find the points of sine or cosine from the given angle of $\theta ={60}^{\circ }$

Answer to Problem 7SP

  $\begin{array}{l}\sin \left({60}^{\circ }\right)=\frac{\sqrt{3}}{2}\\ \cos \left({60}^{\circ }\right)=\frac{1}{2}\end{array}$

Explanation of Solution

Given information: $\theta ={60}^{\circ }$

Calculation:

Let us convert the angles from degree to radian by using the following formula

  $\theta ={\theta }_{\mathrm{deg}ree}\times \frac{\pi }{180}$

Consider the angle in degree

  $\begin{array}{l}{\theta }_{\mathrm{deg}ree}={60}^{\circ }\\ {\theta }_{radian}=60\times \frac{\pi }{180}\\ =\frac{\pi }{3}\end{array}$

Let us calculate sine and cosine of the angle.

  $\begin{array}{l}\sin \left({60}^{\circ }\right)=\frac{\sqrt{3}}{2}\\ \cos \left({60}^{\circ }\right)=\frac{1}{2}\end{array}$

b

To determine

To find the points of sine or cosine from the given angle of $\theta =-{60}^{\circ }$

Answer to Problem 7SP

  $\begin{array}{l}\sin \left(-{60}^{\circ }\right)=-\frac{\sqrt{3}}{2}\\ \cos \left(-{60}^{\circ }\right)=\frac{1}{2}\end{array}$

Explanation of Solution

Given information: $\theta =-{60}^{\circ }$

Calculation:

Consider the angle in degree

  $\begin{array}{l}{\theta }_{\mathrm{deg}ree}=-{60}^{\circ }\\ {\theta }_{radian}=-60\times \frac{\pi }{180}\\ =-\frac{\pi }{3}\end{array}$

Let us calculate sine and cosine of the angle.

  $\begin{array}{l}\sin \left(-{60}^{\circ }\right)=-\frac{\sqrt{3}}{2}\\ \cos \left(-{60}^{\circ }\right)=\frac{1}{2}\end{array}$

c

To determine

To find the points of sine or cosine from the given angle of $\theta ={110}^{\circ }$

Answer to Problem 7SP

  $\begin{array}{l}\sin \left({110}^{\circ }\right)=0.9397\\ \cos \left({110}^{\circ }\right)=-0.3420\end{array}$

Explanation of Solution

Given information: $\theta ={110}^{\circ }$

Calculation:

Consider the angle in degree

  $\begin{array}{l}{\theta }_{\mathrm{deg}ree}={110}^{\circ }\\ {\theta }_{radian}=110\times \frac{\pi }{180}\\ =\frac{11\pi }{18}\\ =1.9198\end{array}$

Let us calculate sine and cosine of the angle.

  $\begin{array}{l}\sin \left({110}^{\circ }\right)=0.9397\\ \cos \left({110}^{\circ }\right)=-0.3420\end{array}$

d

To determine

To find the points of sine or cosine from the given angle of $\theta =-{190}^{\circ }$

Answer to Problem 7SP

  $\begin{array}{l}\sin \left(-{190}^{\circ }\right)=0.1736\\ \cos \left(-{190}^{\circ }\right)=-0.9848\end{array}$

Explanation of Solution

Given information: $\theta =-{190}^{\circ }$

Calculation:

Consider the angle in degree

  $\begin{array}{l}{\theta }_{\mathrm{deg}ree}=-{190}^{\circ }\\ {\theta }_{radian}=-190\times \frac{\pi }{180}\\ =-\frac{19\pi }{18}\\ =-3.3161\end{array}$

Let us calculate sine and cosine of the angle.

  $\begin{array}{l}\sin \left(-{190}^{\circ }\right)=0.1736\\ \cos \left(-{190}^{\circ }\right)=-0.9848\end{array}$

e

To determine

To find the points of sine or cosine from the given angle of $\theta ={1160}^{\circ }$

Answer to Problem 7SP

  $\begin{array}{l}\sin \left({1160}^{\circ }\right)=0.9848\\ \cos \left({1160}^{\circ }\right)=0.1736\end{array}$

Explanation of Solution

Given information: $\theta ={1160}^{\circ }$

Calculation:

Consider the angle in degree

  $\begin{array}{l}{\theta }_{\mathrm{deg}ree}={1160}^{\circ }\\ {\theta }_{radian}=1160\times \frac{\pi }{180}\\ =-\frac{58\pi }{9}\\ =20.2458\end{array}$

Let us calculate sine and cosine of the angle.

  $\begin{array}{l}\sin \left({1160}^{\circ }\right)=0.9848\\ \cos \left({1160}^{\circ }\right)=0.1736\end{array}$

Graph of phase-plane with the angles is shown below

  

Sine graph with the sine of given angles is plotted as shown below

  

Cosine graph with the cosine of given angles is plotted as shown below