Chapter 4.3, Problem 51E

(a)

To determine

To calculate: The value of $\tan 60°$ with the help of the trigonometric identities.

Answer to Problem 51E

The value of $\tan 60°$ is $\sqrt{3}$ .

Explanation of Solution

Given Information:

The value of $\sin 60°=\frac{\sqrt{3}}{2}$ and $\cos 60°=\frac{1}{2}$ .

Formula Used:

Trigonometric identity $\tan \theta =\frac{\sin \theta }{\cos \theta }$

Calculation:

Substitute $60°$ for $\theta$ in the identity $\tan \theta =\frac{\sin \theta }{\cos \theta }$ .

  $\tan 60°=\frac{\sin 60°}{\cos 60°}$

Substitute $\frac{\sqrt{3}}{2}$ for $\sin 60°$ and $\frac{1}{2}$ for $\cos 60°$ in above identity.

  $\begin{array}{c}\tan 60°=\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}\\ =\sqrt{3}\end{array}$

Therefore,the value of $\tan 60°$ is $\sqrt{3}$ .

(b)

To determine

To calculate: The value of $\sin 30°$ with the help of the trigonometric identities.

Answer to Problem 51E

The value of $\sin 30°$ is $\frac{1}{2}$ .

Explanation of Solution

Given Information:

The value of $\sin 60°=\frac{\sqrt{3}}{2}$ and $\cos 60°=\frac{1}{2}$ .

Formula Used:

Trigonometric identity $\cos \left(90°-\theta \right)=\sin \theta$ .

Calculation:

Substitute $30°$ for $\theta$ in the above identity.

  $\begin{array}{c}\cos \left(90°-30°\right)=\sin 30°\\ \cos 60°=\sin 30°\end{array}$

Substitute $\frac{1}{2}$ for $\cos 60°$ .

  $\begin{array}{c}\cos 60°=\sin 30°\\ \frac{1}{2}=\sin 30°\end{array}$

Therefore the value of $\sin 30°$ is $\frac{1}{2}$ .

(c)

To determine

To calculate: The value of $\cos 30°$ with the help of the trigonometric identities.

Answer to Problem 51E

The value of $\cos 30°$ is $\frac{\sqrt{3}}{2}$ .

Explanation of Solution

Given Information:

The value of $\sin 60°=\frac{\sqrt{3}}{2}$ and $\cos 60°=\frac{1}{2}$ .

Formula Used:

Trigonometric identity $\sin \left(90°-\theta \right)=\cos \theta$

Calculation:

Substitute $30°$ for $\theta$ in the above identity.

  $\sin \left(90°-30°\right)=\cos 30°$ , this gives that $\cos 30°=\sin 60°$ .

Substitute $\frac{\sqrt{3}}{2}$ for $\sin 60°$ .

  $\begin{array}{c}\cos 30°=\sin 60°\\ =\frac{\sqrt{3}}{2}\end{array}$

Therefore, the value of $\cos 30°$ is $\frac{\sqrt{3}}{2}$ .

(d)

To determine

The value of $\cot 60°$ with the help of the trigonometric identities.

Answer to Problem 51E

The value of $\cot 60°$ is $\frac{\sqrt{3}}{3}$ .

Explanation of Solution

Given Information:

The value of $\sin 60°=\frac{\sqrt{3}}{2}$ and $\cos 60°=\frac{1}{2}$ .

Formula Used:

Trigonometric identity $\cot \theta =\frac{\cos \theta }{\sin \theta }$

Calculation:

Substitute $60°$ for $\theta$ in the above identity.

  $\cot 60°=\frac{\cos 60°}{\sin 60°}$ .

Substitute $\frac{\sqrt{3}}{2}$ for $\sin 60°$ and $\frac{1}{2}$ for $\cos 60°$ .

  $\begin{array}{c}\cot 60°=\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}\\ =\frac{1}{\sqrt{3}}\\ =\frac{\sqrt{3}}{3}\end{array}$

Therefore, the value of $\cot 60°$ is $\frac{\sqrt{3}}{3}$ .