Trigonometric equations can be solved in degrees or radians using CAST and its period to find other solutions within the range, including multiple or compound angles and the wave function.
Part of MathsAlgebraic and trigonometric skills
Solve \({\tan ^2}x = 3\), where \(0 \le x \le 360\).
\({\tan ^2}x = 3\)
\(\tan x = \pm \sqrt 3\)
Since this is tan and can be positive or negative, then this means we are in all four quadrants.
\(\tan x = \sqrt 3\)
\(x = {\tan ^{ - 1}}(\sqrt 3 )\)
\(x = 60^\circ\)
\(x = 180^\circ - 60^\circ\)
\(x = 120^\circ\)
\(x = 180^\circ + 60^\circ\)
\(x = 240^\circ\)
\(x = 360^\circ - 60^\circ\)
\(x = 300^\circ\)
Therefore \(x^\circ = 60^\circ ,\,120^\circ ,\,240^\circ ,\,300^\circ\)
Solve \(8{\sin ^2}x^\circ + 2\sin x^\circ - 3 = 0\), where \(0 \le x \le 360\).
First we need to factorise the quadratic equation. To make this easier, change 'sin' to 's'.
\(8{s^2} + 2s - 3 = 0\)
\((4s + 3)(2s - 1) = 0\)
4s + 3 = 0
2s - 1 = 0
Therefore \(x = 30^\circ ,\,150^\circ ,\,228.6^\circ ,\,311.4^\circ\)