Solve the equation (x in radians and in degrees) for all exact solutions where appropriate. Round approximate answers
in radians to four decimal places and approximate answers in degrees to the nearest tenth.
-$$\frac { 4 \tan \theta } { 5 - \tan ^ { 2 } \theta } = 1$$
A) $\varnothing$
B) $\left\{ 0 ^ { \circ } + 180 ^ { \circ } \mathrm { n } , 45 ^ { \circ } + 180 ^ { \circ } \mathrm { n } , 101.3 ^ { \circ } + 180 ^ { \circ } \mathrm { n } \right\}$
C) $\left\{ 45 ^ { \circ } + 360 ^ { \circ } \mathrm { n } , 101.3 ^ { \circ } + 360 ^ { \circ } \mathrm { n } \right\}$
D) $\left\{ 45 ^ { \circ } + 180 ^ { \circ } \mathrm { n } , 101.3 ^ { \circ } + 180 ^ { \circ } \mathrm { n } \right\}$

Question

Quizplus · Accepted Answer

To solve the given equation $\frac { 4 \tan \theta } { 5 - \tan ^ { 2 } \theta } = 1$, we first rearrange it to find the values of $\tan \theta$. Multiplying both sides by $5 - \tan^2 \theta$ and then moving all terms to one side gives a quadratic equation in $\tan \theta$. Solving this quadratic equation yields two solutions for $\tan \theta$, which correspond to angles $\theta = 45^\circ$ and an angle slightly larger than $90^\circ$, which when calculated accurately gives $\theta \approx 101.3^\circ$. These solutions repeat every $180^\circ$ due to the periodicity of the tangent function, leading to the set of solutions $\left\{ 45 ^ { \circ } + 180 ^ { \circ } \mathrm { n } , 101.3 ^ { \circ } + 180 ^ { \circ } \mathrm { n } \right\}$.

Solve the Equation (X in Radians and in Degrees) for All