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

Answer:

$\frac{\pi}{6},\text{ }\frac{5}{6}\pi$

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:

$\begin{gathered} 3y^2\text{ = 1} \\ y^2\text{ = }\frac{1}{3} \\ \\ y\text{ = }\sqrt[]{\frac{1}{3}} \\ \\ y\text{ = }\pm\sqrt[]{\frac{1}{3}} \\ \\ or\text{ } \\ \\ y\text{ = }\frac{\pm\sqrt[]{3}}{3} \end{gathered}$

Recall that we made a substitution for tan x:

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

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