Trigonometric Equations Reducible to Quadratic for Math Skills
Trigonometric Equations Reducible to Quadratic for Math Skills are based on trigonometric identities such as;
$$
\sin^2{x} + \cos^2{x} = 1 \\
1 + \cot^2{x} = \csc^2{x} \\
\tan^2{x} + 1 = \sec^2{x} \\
$$
Practice Questions
Question 1
Solve \( 2 \cos^2{x} – 3 \cos{x} + 1 = 0 \) for \( 0^\circ \ge x \ge 360^\circ \).
\( \begin{aligned} \displaystyle
(\cos{x}-1)(2 \cos{x}-1) &= 0 \\
\cos{x} &= 1 \text{ or } \cos{x} = \frac{1}{2} \\
x &= 0^\circ, 360^\circ \text{ or } x = 60^\circ, 360^\circ – 60^\circ \\
\therefore x &= 0^\circ, 360^\circ, 60^\circ, 300^\circ \\
\end{aligned} \\ \)
Question 2
Solve \( \sec^2{x} + 2 \tan{x} = 0 \) for \( 0^\circ \ge x \ge 360^\circ \).
\( \begin{aligned} \displaystyle
(1 + \tan^2{x}) + 2 \tan{x} &= 0 \\
tan^2{x} + 2 \tan{x} +1 &= 0 \\
(\tan{x} + 1)^2 &= 0 \\
\tan{x} &= -1 \\
x &= 180^\circ – 45^\circ, 360^\circ – 45^\circ \\
\therefore x &= 135^\circ, 315^\circ \\
\end{aligned} \\ \)
Question 3
Solve \( \cot^2{x} = \csc{x} + 1 \) for \( 0^\circ \ge x \ge 360^\circ \).
\( \begin{aligned} \displaystyle
\csc^2{x} – 1 &= \csc{x} +1 \\
\csc^2{x} – \csc{x} – 2 &= 0 \\
(\csc{x} – 2)(\csc{x} + 1) &= 0 \\
\csc{x} &= 2 \text{ or } \csc{x} = -1 \\
\sin{x} &= \frac{1}{2} \text{ or } \sin{x} = -1 \\
x &= 30^\circ, 180^\circ-30^\circ \text{ or } x = 270^\circ \\
\therefore x &= 30^\circ, 150^\circ, 270^\circ \\
\end{aligned} \\ \)
Question 4
Solve \( \cos^2{x} + \cos{x} = \sin^2{x} \) for \( 0^\circ \ge x \ge 360^\circ \).
\( \begin{aligned} \displaystyle
\cos^2{x} + \cos{x} &= 1 – \cos^2{x} \\
2 \cos^2{x} + \cos{x} – 1 &= 0 \\
(2 \cos{x} – 1)(\cos{x} + 1) &= 0 \\
\cos{x} &= \frac{1}{2} \text{ or } \cos{x} = -1 \\
x &= 60^\circ, 360^\circ – 60^\circ \text{ or } x = 180^\circ \\
\therefore x &= 60^\circ, 300^\circ, 180^\circ \\
\end{aligned} \\ \)
Related Topics
Algebra Equations Reducible to Quadratic
Fraction Equations Reducible to Quadratic
Equations Reducible to Quadratic by Substitution
Exponential Equations Reducible to Quadratic
Logarithmic Equations Reducible to Quadratic
Surd Equations Reducible to Quadratic
