Chapter 6, Problem 1CLT

a.

To determine

Sketch the angle in standard position.

Answer to Problem 1CLT

  

Explanation of Solution

Given information:

Consider the angle $\theta =-{120}^{\circ }$ .

Sketch the angle in standard position.

Calculation:

Consider the angle $\theta =-{120}^{\circ }$ .

Draw the angle as shown below,

  

Hence, the angle has been sketched in the standard form.

b.

To determine

Answer to Problem 1CLT

  $\boxed{{240}^{\circ }}$

Explanation of Solution

Given information:

Consider the angle $\theta =-{120}^{\circ }$ .

Determine the coterminal angle in the interval $[0^{\circ },{360}^{\circ })$ .

Calculation:

Since, $-{120}^{\circ }$ does not lies in the given interval, the coterminal angle for the given angle is ${240}^{\circ }$ which lies in the given interval has the same values of trigonometric functions as that for given angle.

Hence, , the coterminal angle for the given angle is $\boxed{{240}^{\circ }}$ .

c.

To determine

Rewrite the angle in radian measure.

Answer to Problem 1CLT

  $=\boxed{\frac{-2\pi }{3}}$

Explanation of Solution

Given information: Consider the angle $\theta =-{120}^{\circ }$ .

Rewrite the angle in radian measure as multiple of $\pi$ .

Calculation:

The angle in degree measure can be converted to radian measure as follows:

  $\frac{\theta }{{180}^{\circ }}\times \pi =\frac{-{120}^{\circ }}{{180}^{\circ }}\times \pi$

  $=\boxed{\frac{-2\pi }{3}}$ .

Hence, $\theta$$=\boxed{\frac{-2\pi }{3}}$ .

d.

To determine

Find the reference angle $\theta \text{'}$

Answer to Problem 1CLT

  $\boxed{{60}^{\circ }}$

Explanation of Solution

Given information: Consider the angle $\theta =-{120}^{\circ }$ .

Find the reference angle $\theta \text{'}$

Calculation:

The reference angle to the given angle is ${60}^{\circ }$ .

Hence, $\theta \text{'}$ = $\boxed{{60}^{\circ }}$

e.

To determine

Answer to Problem 1CLT

  $\boxed{\sin \theta =-\frac{\sqrt{3}}{2}}$

  $\boxed{\cos \theta =-\frac{1}{2}}$

  $\boxed{\tan \theta =\sqrt{3}}$

  $\boxed{\sec \theta =-2}$

  $\boxed{\csc \theta =-\frac{2}{\sqrt{3}}}$

  $\boxed{\cot \theta =\frac{1}{\sqrt{3}}}$

Explanation of Solution

Given information: Consider the angle $\theta =-{120}^{\circ }$ .

Find th exact values of the six trigonometric functions of $\theta$ .

Calculation:

The values of the trigonometric functions have been calculated below:

  $\begin{array}{l}\sin \theta =\sin -{120}^{\circ }\\ \sin \theta =-\sin {120}^{\circ }\\ \boxed{\sin \theta =-\frac{\sqrt{3}}{2}}\end{array}$

  $\begin{array}{l}\cos \theta =\sin -{120}^{\circ }\\ \cos \theta =\cos {120}^{\circ }\\ \boxed{\cos \theta =-\frac{1}{2}}\end{array}$

  $\begin{array}{l}\tan \theta =\frac{\sin \theta }{\cos \theta }\\ \tan \theta =\frac{\sin -{120}^{\circ }}{\cos -{120}^{\circ }}\\ \boxed{\tan \theta =\sqrt{3}}\end{array}$

  $\begin{array}{l}\sec \theta =\frac{1}{\cos \theta }\\ \sec \theta =\frac{1}{\cos -{120}^{\circ }}\\ \boxed{\sec \theta =-2}\end{array}$

  $\begin{array}{l}\csc \theta =\frac{1}{\sin \theta }\\ \csc \theta =\frac{1}{\sin -{120}^{\circ }}\\ \boxed{\csc \theta =-\frac{2}{\sqrt{3}}}\end{array}$

  $\begin{array}{l}\cot \theta =\frac{\cos \theta }{\sin \theta }\\ \cot \theta =\frac{\cos -{120}^{\circ }}{\sin -{120}^{\circ }}\\ \boxed{\cot \theta =\frac{1}{\sqrt{3}}}\end{array}$

Hence, the result is calculated.