$\tan \left(\frac{\pi}{4}x\right)+\sqrt{3}=0$

View step-by-step help

Home

Math Problems

Algebra 2

Simplify expressions using trigonometric identities

Full solution

\tan \left(\frac{\pi}{4}x\right)+\sqrt{3}=0

Isolate tangent term: Express the equation in terms of $\tan\left(\frac{\pi}{4}x\right)$ by subtracting $\sqrt{3}$ from both sides to isolate the tangent term. $\newline$ $\tan\left(\frac{\pi}{4}x\right) + \sqrt{3} = 0$ $\newline$ $\tan\left(\frac{\pi}{4}x\right) = -\sqrt{3}$
Identify tangent values: Recognize that the tangent of an angle is $-\sqrt{3}$ at $240$ degrees (or $\frac{4\pi}{3}$ radians) and $300$ degrees (or $\frac{5\pi}{3}$ radians) in the unit circle, which correspond to angles in the second and third quadrants where tangent is negative. Since the argument of the tangent function is $(\frac{\pi}{4})x$ , we need to find the values of $x$ that make $(\frac{\pi}{4})x$ equal to $\frac{4\pi}{3}$ or $\frac{5\pi}{3}$ .
Solve for x ( $1$ st equation): Set up the first equation for $(\pi/4)x = 4\pi/3$ and solve for x. $\newline$ $(\pi/4)x = 4\pi/3$ $\newline$ Multiply both sides by $4/\pi$ to solve for x. $\newline$ $x = (4\pi/3) \cdot (4/\pi)$ $\newline$ $x = 16/3$
Solve for x ( $2$ nd equation): Set up the second equation for $(\pi/4)x = 5\pi/3$ and solve for x. $(\pi/4)x = 5\pi/3$ Multiply both sides by $4/\pi$ to solve for x. $x = (5\pi/3) \cdot (4/\pi)$ $x = 20/3$
Consider periodicity: However, we must consider the periodicity of the tangent function, which is $\pi$ . Since the argument of the tangent function is $(\pi/4)x$ , the period in terms of $x$ is $4$ . This means that we can add or subtract multiples of $4$ to our solutions to find other solutions.
General solution for $x$ : The general solution for $x$ is then given by: $\newline$ $x = \frac{16}{3} + 4k$ or $x = \frac{20}{3} + 4k$ , where $k$ is any integer.