Simultaneous Trigonometric Equations - Practice Questions & MCQ

Simultaneous Trigonometric Equations

Sometimes the value of angle(s) which satisfies more than one trigonometric equation simultaneously is asked. We call such a system of equations as Simultaneous Trigonometric Equations.

Illustration

General value of $x$ which satisfies the equation $\cos x=-1 / 2$ and $\cot x=1 / \sqrt{3}$
First, find the value of $x$ in $[0,2 \pi)$ which satisfies two given equations separately

$
\begin{aligned}
& \cos x=-1 / 2 \Rightarrow x=2 \pi / 3 \text { and } 4 \pi / 3 \\
& \cot x=1 / \sqrt{ } 3 \Rightarrow x=\pi / 3 \text { and } 4 \pi / 3
\end{aligned}
$
Now select the value x which satisfies both the equation
Here common value is $4 \pi / 3$
Also, we know that $\cos$ and $\cos$ functions will repeat after an interval of $2 \pi$, so the required general solution is $x=2 n \pi+4 \pi / 3$