Determine all solutions of the equation in radians.
-Find $\tan \frac { x } { 2 }$, given that $\tan x = - 3$ and $x$ terminates in $90 ^ { \circ }

Question

Determine all solutions of the equation in radians.
-Find $\tan \frac { x } { 2 }$, given that $\tan x = - 3$ and $x$ terminates in $90 ^ { \circ } < x < 180 ^ { \circ }$.

Quizplus · Accepted Answer

The half-angle formula for tangent is $\tan\left(\frac{x}{2}\right) = \frac{1 - \cos(x)}{\sin(x)}$. Given $\tan(x) = -3$, we can find $\sin(x)$ and $\cos(x)$ using the identity $\tan(x) = \frac{\sin(x)}{\cos(x)}$. Since $\tan(x) = -3$, and knowing that $\sin^2(x) + \cos^2(x) = 1$, we can solve for $\sin(x)$ and $\cos(x)$ in the given quadrant (II, where $\sin(x) > 0$ and $\cos(x) < 0$). This gives us $\sin(x) = \frac{3}{\sqrt{10}}$ and $\cos(x) = -\frac{1}{\sqrt{10}}$. Substituting these into the half-angle formula and simplifying gives $\tan\left(\frac{x}{2}\right) = \frac{\sqrt{10} + 1}{3}$.

Determine All Solutions of the Equation in Radians tan⁡x2\tan \frac { x } { 2 }tan2x​

Determine All Solutions of the Equation in Radians $\tan \frac { x } { 2 }$