Equations and Identities
Quadratics
- It is possible to have a quadratic trigonometric equation.
- When this happens, you need to substitute in \(x\) into the equation instead of the sin/cos/tan value.
\(\tan^2\theta=4\) in the interval \(0 \leq x\leq 360^\circ\)
- \(x^2=4\)
- \(x=2\)
- \(\tan\theta=2\)
- \(\tan^{-1}(2)=63.43^\circ\)
- \(x_{1}=63.43^\circ\)
- \(x_{2}=243.4^\circ\)
- \(x_{3}=116.6^\circ\)
- \(x_{4}=296.6^\circ\)
\(2\sin^2x+5\sin x-3=0\)
- Sub in x
- \(2x^2+5x-3=0\)
- Solutions are \(\frac{1}{2}\) and \(-3\)
- \(\sin x=\frac{1}{2}\) and \(\sin x=-3\)
- \(\sin^{-1} \frac{1}{2} = 30\) and \(\sin^{-1} -3\) has no solutions.
- Using cast diagram, solutions are \(30^\circ\) and \(150^\circ\)