Further trigonometric equations
Example 1
Solve \({\tan ^2}x = 3\), where \(0 \le x \le 360\).
Solution
\({\tan ^2}x = 3\)
\(\tan x = \pm \sqrt 3\)
Since this is tan and can be positive or negative, then this means we are in all four quadrants.
First quadrant
\(\tan x = \sqrt 3\)
\(x = {\tan ^{ - 1}}(\sqrt 3 )\)
\(x = 60^\circ\)
Second quadrant
\(x = 180^\circ - 60^\circ\)
\(x = 120^\circ\)
Third quadrant
\(x = 180^\circ + 60^\circ\)
\(x = 240^\circ\)
Fourth quadrant
\(x = 360^\circ - 60^\circ\)
\(x = 300^\circ\)
Therefore \(x^\circ = 60^\circ ,\,120^\circ ,\,240^\circ ,\,300^\circ\)
Example 2
Solve \(8{\sin ^2}x^\circ + 2\sin x^\circ - 3 = 0\), where \(0 \le x \le 360\).
Solution
First we need to factorise the quadratic equation. To make this easier, change 'sin' to 's'.
\(8{s^2} + 2s - 3 = 0\)
\((4s + 3)(2s - 1) = 0\)
Therefore \(x = 30^\circ ,\,150^\circ ,\,228.6^\circ ,\,311.4^\circ\)