Chapter 5, Problem 15RE

(a)

To determine

To find: Theexact value of trigonometric function if not possible use a calculator and correct up-to five decimal places.

Answer to Problem 15RE

The exact value is $-0.57735$ .

Explanation of Solution

Given: $\mathrm{Tan}\frac{\text{5π}}{6}$

Concept used:

  ${\text{πradian=180}}^{\text{0}}$ .

  $\text{1radian=}\frac{{\text{180}}^{\text{0}}}{\text{π}}$ .

In second quadrant $\text{Sinθ}$ is positive.

Since, its changing horizontally the trigonometric doesn’t change $\text{Sinθ}$ to $\text{Cosθ}$ it remains same.

Calculation:

  $\mathrm{Tan}\frac{\text{5π}}{6}$ = $\mathrm{Tan}\left({150}^0\right)$ .

so, the angle lies in 2^nd quadrant.

Actually, the angle can be measured directly by unit circle, let’s solved it without the use of unit circle.

  $\mathrm{Tan}\left({150}^0\right)$ can be written as $\mathrm{Tan}\left(\text{π-}\frac{\text{π}}{6}\right)$

  $\mathrm{Tan}\left(\text{π-}\frac{\text{π}}{6}\right)\text{=-Tan}\frac{\text{π}}{6}=-\frac{1}{\sqrt{3}}.$

Hence the exact value is $-0.57735$ .

(b)

To determine

To find:The exact value of trigonometric function if not possible use a calculator and correct up-to five decimal places.

Answer to Problem 15RE

The exact value is $-1.73205$ .

Explanation of Solution

Given: $\text{Cot}\frac{\text{5π}}{\text{6}}$ .

Concept used:In second quadrant $\text{Cotθ}$ is negative.

Since, its changing horizontally the trigonometric doesn’t change $\text{Cosθ}$ to $\text{Sinθ}$ it remains same.

Calculation:

  $\text{Cot}\frac{\text{5π}}{\text{6}}$ = $\text{Cot}\left({\text{150}}^{\text{0}}\right)$ so, the angle lies in 2^nd quadrant.

Actually, the angle can be measured directly by unit circle, let’s solved it without the use of unit circle.

  $\text{Cot}\left({\text{150}}^{\text{0}}\right)$

  $\text{Cot}\frac{\text{5π}}{\text{6}}$ = $\text{Cot}\left({\text{150}}^{\text{0}}\right)\text{=Cot}\left(\text{π-}\frac{\text{π}}{\text{6}}\right)$

  $\Rightarrow -\text{Cot}\frac{\text{π}}{\text{6}}=-\sqrt{3}.$

Hence the exact value is $-1.73205$ .