Question

Use trigonometric identities and algebraic methods, as necessary, to solve the following trigonometric equation. Please identify all possible solutions by including allanswers in [0, x) and indicating the remaining answers by using n to represent any integer. Round your answer to four decimal places, if necessary. If there is no solution,indicate "No Solution."he3tan(x) + 1 = 0

Answer 1

`(\pi)/(6),\text{ }(5)/(6)\pi`

Step-by-step explanation:

Here, we want to solve the given equation

`\text{Let y = tan x}`

This simply means that:

`\begin{gathered} \sqrt[]{3\text{ }}y+1\text{ = 0} \\ \sqrt[]{3}\text{ y = -1} \\ \end{gathered}`

From here, we can square both sides:

`\begin{gathered} 3y^2\text{ = 1} \\ y^2\text{ = }(1)/(3) \\ \\ y\text{ = }\sqrt[]{(1)/(3)} \\ \\ y\text{ = }\pm\sqrt[]{(1)/(3)} \\ \\ or\text{ } \\ \\ y\text{ = }\frac{\pm\sqrt[]{3}}{3} \end{gathered}`

Recall that we made a substitution for tan x:

`\begin{gathered} \frac{\sqrt[]{3}}{3}\text{ = tan x} \\ x\text{ = }\tan ^(-1)(\frac{\sqrt[]{3}}{3}) \\ x\text{ = 30 deg = }(\pi)/(6) \end{gathered}`

Let us find the other values of x between 0 and pi

We know that tan is negative on the second quadrant

We have the values in this quadrant as 180-theta

Which is 180-30 = 150 degrees which is same as 150/180 pi = 5/6pi