Chapter 7.3, Problem 48AYU

In Problems 47-58, use a calculator to solve each equation on the interval $0\le \theta <2\pi$. Round answers to two decimal places.

$\cot \theta =2$

To determine

To find: The trigonometric equation $\text{cot }\theta =2$ using the calculator, and rounding off the answer to two decimal places, $0\le \theta <2\pi$.

Answer to Problem 48AYU

The solution set for the trigonometric equation $\text{cot }\theta =2=\left\{{0.46}^c,{5.80}^c\right\},0\le \theta <2\pi$.

Explanation of Solution

Given:

Trigonometric equation $\text{cot }\theta =2$, $0\le \theta <2\pi$.

Calculation:

To solve $\text{cot }\theta =2$ on a calculator, the mode has to be kept in radians mode. But it is not possible to calculate $\text{cot }\theta$ on the calculator, directly. Therefore, to solve for $\theta$ we need to calculate its $\text{cos}$ equivalent.

$\text{cot }\theta =2=x/y$

Equation of the circle $=x^2+y^2=r^2$.

Using the formula,

$r=\sqrt{\left(x^2+y^2\right)}$

$r=\sqrt{\left(2^2+1^2\right)}=\sqrt{5}$

$\text{cos }\theta =2/\sqrt{5}=0.8944$

$\Theta ={\text{ cos}}^{-1}\left(0.8944\right)\approx 0.463$

Rounding off to two decimal places,

$\Theta ={\text{ cos}}^{-1}\left(0.8944\right)=0.46$ radians.

By definition,

$\theta ={\text{ cos}}^{-1}x$, the angle $\theta$ is defined within the range as $0\le \theta \le \pi$.

For $\text{cos }\theta =0.8944$, we need to find solutions for $\theta$ for $0\le \theta <2\pi$ interval.

$\text{cos }\theta =0.8944>0$

Therefore,

$\theta$ lies in I and IV quadrants.

I quadrant angle $\theta ={0.46}^c$.

IV quadrant angle $\theta =\text{ 2}\pi -0.46=\text{ 5}.80$.

The solution set for Trigonometric equation $\text{cot }\theta =2={0.46}^c,{5.80}^c$.