Question

Explain in your own words how you would use your calculator to solve the equation $\cos x=-0.6,0\le x\le 2\mathrm{\pi }$. How would you modify your approach to solve the equation $\cot x=-0.6,0\le x\le 2\mathrm{\pi }$?

Short answer

The solution set of$\cos x=-0.6\mathrm{is}\left\{2.21,4.07\right\}$ and$\cot x=-0.6\mathrm{is}\left\{0.197,3.337\right\}.$

Step-by-step solution

Step 1. Given Information 

Explain in your own words how you would use your calculator to solve the equation $\cos x=-0.6,0\le x\le 2\mathrm{\pi }$. How would you modify your approach to solve the equation $\cot x=-0.6,0\le x\le 2\mathrm{\pi }$?

Step 2. To solve cos x=-0.6, 0≤x≤2πon a calculator,

First set the mode to radians. Then use the ${\cos }^{-1}$key to obtain

$x={\cos }^{-1}\left(-0.6\right)$

$x\approx 2.214$

Rounded to two decimal places, $2.21$radian.

Step 3. Since cosine function is positive in fourth quadrant so we can use the identify cos2π-θ=cosθ

To find the second solution within $[0,2\mathrm{\pi }]$

$$\begin{gathered} x=2\mathrm{\pi }-2.21 \\ \mathrm{x}=4.07 \end{gathered}$$

Then the solution set isrole="math" localid="1647109842173" $\left\{2.21,4.07\right\}$.

Step 4. To solve cotx=-0.6 on calsulator.

We firstly write $\cot x=\frac{1}{\tan x}$

Then the equation is

$$\begin{gathered} \frac{1}{\tan x}=5 \\ \frac{1}{\tan x}·\tan x=5·\tan x \\ 1=5·\tan x \\ \frac{1}{5}=\frac{5}{5}·\tan x \\ \tan x=0.2 \end{gathered}$$

use the ${\cot }^{-1}$key to obtain.

Step 5. Now First set the mode to radians. Then use the tan-1 key to obtain.

$$\begin{gathered} x={\tan }^{-1}0.2 \\ x\approx 0.197 \end{gathered}$$

Rounded to two decimal places, $0.19$radian.

Step 6. Since tangent function is positive in third quadrant so we can use the identify tanπ+θ=tanθ

To find the second solution within$[0,2\mathrm{\pi }]$

$$\begin{gathered} x=\mathrm{\pi }+0.197 \\ \mathrm{x}=3.337 \end{gathered}$$

Then the solution set is$\left\{0.197,3.337\right\}$