Chapter 13.6, Problem 34PPSE

Solution

To solve: The equation $\tan \left(\theta \right)=-2$ in the interval from $0\text{ to }2\pi$ and round the answer to the nearest hundredth.

The solution of the equation $\tan \left(\theta \right)=-2$ in the interval $\left(0,2\pi \right)$ is $\theta =2.03\text{ and }5.18$ .

Given:

The equation is $\tan \left(\theta \right)=2$ over the interval $\left(0,2\pi \right)$ .

Concept used:

Calculation:

Sketch the graph of the tangent curve $\tan \left(\theta \right)$ and the line $y=-2$ separately as shown below and look for intersecting points in the interval $\left(0,2\pi \right)$ . Therefore, from the graph of the function shown below.

  

The graph of the curves $y=-2$ and tangent curve $\tan \left(\theta \right)$ , it can be interpreted that the curve cut at the points $\left(2.034,-2\right)\text{ and }\left(5.176,-2\right)$ respectively in the interval $\left(0,2\pi \right)$ . So, approximated points are $\left(2.03,-2\right)\text{ and }\left(5.18,-2\right)$ to nearest hundredths and write the $\theta -$ values as the solution of the equation $\tan \left(\theta \right)=-2$ .

Hence the solution of the equation $\tan \left(\theta \right)=-2$ in the interval $\left(0,2\pi \right)$ is $\theta =2.03\text{ and }5.18$ .

Conclusion:

The solution of the equation $\tan \left(\theta \right)=-2$ in the interval $\left(0,2\pi \right)$ is $\theta =2.03\text{ and }5.18$ .