# Trigonometric Equations Reducible to Quadratics

Trigonometric Equations Reducible to Quadratic for Math Skills are based on trigonometric identities such as;
\large \begin{align} \sin^2{x} + \cos^2{x} &= 1 \\ 1 + \cot^2{x} &= \csc^2{x} \\ \tan^2{x} + 1 &= \sec^2{x} \ \end{align}

### Question 1

Solve $2 \cos^2{x}-3 \cos{x} + 1 = 0$ for $0^\circ \le x \le 360^\circ$.

\begin{aligned} \displaystyle (\cos{x}-1)(2 \cos{x}-1) &= 0 \\ \cos{x} &= 1 \text{ or } \cos{x} = \frac{1}{2} \\ x &= 0^\circ, 360^\circ \text{ or } x = 60^\circ, 360^\circ-60^\circ \\ \therefore x &= 0^\circ, 360^\circ, 60^\circ, 300^\circ \end{aligned}

### Question 2

Solve $\sec^2{x} + 2 \tan{x} = 0$ for $0^\circ \le x \le 360^\circ$.

\begin{aligned} \displaystyle (1 + \tan^2{x}) + 2 \tan{x} &= 0 \\ \tan^2{x} + 2 \tan{x} +1 &= 0 \\ (\tan{x} + 1)^2 &= 0 \\ \tan{x} &= -1 \\ x &= 180^\circ-45^\circ, 360^\circ-45^\circ \\ \therefore x &= 135^\circ, 315^\circ \end{aligned}

### Question 3

Solve $\cot^2{x} = \csc{x} + 1$ for $0^\circ \le x \le 360^\circ$. Note $\csc x= \text{cosec } x$

\begin{aligned} \displaystyle \csc^2{x}-1 &= \csc{x} +1 \\ \csc^2{x}-\csc{x}-2 &= 0 \\ (\csc{x}-2)(\csc{x} + 1) &= 0 \\ \csc{x} &= 2 \text{ or } \csc{x} = -1 \\ \sin{x} &= \frac{1}{2} \text{ or } \sin{x} = -1 \\ x &= 30^\circ, 180^\circ-30^\circ \text{ or } x = 270^\circ \\ \therefore x &= 30^\circ, 150^\circ, 270^\circ \end{aligned}

### Question 4

Solve $\cos^2{x} + \cos{x} = \sin^2{x}$ for $0^\circ \le x \le 360^\circ$.

\begin{aligned} \displaystyle \cos^2{x} + \cos{x} &= 1-\cos^2{x} \\ 2 \cos^2{x} + \cos{x}-1 &= 0 \\ (2 \cos{x}-1)(\cos{x} + 1) &= 0 \\ \cos{x} &= \frac{1}{2} \text{ or } \cos{x} = -1 \\ x &= 60^\circ, 360^\circ-60^\circ \text{ or } x = 180^\circ \\ \therefore x &= 60^\circ, 300^\circ, 180^\circ \end{aligned}

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