This tutorial explains how to solve trigonometric equations using identities, algebraic manipulation, and the unit circle. Each example includes clear reasoning, graphical interpretation, and complete solutions. The unit circle is especially useful for identifying reference angles and locating all possible solutions.
For additional practice, see: Grade 12 Trigonometric Equations with Detailed Solutions.
Solve the trigonometric equation (find all solutions):
\[ 2\cos x + 2 = 3 \]Start by isolating the trigonometric function.
\[ 2\cos x = 1 \quad \Rightarrow \quad \cos x = \frac{1}{2} \]The cosine function equals \(\frac{1}{2}\) at two angles in the interval \( [0,2\pi) \):
\[ x_1 = \frac{\pi}{3}, \quad x_2 = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3} \]
Since cosine has period \(2\pi\), all solutions are obtained by adding multiples of \(2\pi\):
\[ x = \frac{\pi}{3} + 2k\pi \quad \text{or} \quad x = \frac{5\pi}{3} + 2k\pi, ; k \in \mathbb{Z} \]Conclusion: The equation has infinitely many solutions.
Find all solutions in the interval \( [0,2\pi) \):
\[ -5\cos^2 x + 9\sin x = -3 \]Rewrite \(\cos^2 x\) using the identity \(\cos^2 x = 1 - \sin^2 x\).
\[ -5(1 - \sin^2 x) + 9\sin x = -3 \]Simplify:
\[ 5\sin^2 x + 9\sin x - 2 = 0 \]Let \(u = \sin x\). This transforms the equation into a quadratic.
\[ 5u^2 + 9u - 2 = 0 \]Using the quadratic formula:
\[ u = \frac{-9 \pm \sqrt{121}}{10} \] \[ u_1 = -2 \quad (\text{invalid, since } -1 \le \sin x \le 1) \] \[ u_2 = 0.2 \]Now solve \(\sin x = 0.2\). Sine is positive in Quadrants I and II.
\[
x_1 = \arcsin(0.2), \quad x_2 = \pi - \arcsin(0.2)
\]
Conclusion: There are exactly two solutions in \( [0,2\pi)\).
Find all solutions:
\[ \cot x\cos^2 x = \cot x \]Subtract \(\cot x\) from both sides and factor.
\[ \cot x(\cos^2 x - 1) = 0 \]This gives two equations:
\[ \cot x = 0 \quad \text{or} \quad \cos^2 x - 1 = 0 \]From \(\cot x = 0\):
\[ x = \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z} \]From \(\cos^2 x - 1 = 0\), we get \(\cos x = \pm 1\), which corresponds to \(x = k\pi\). However, \(\cot x\) is undefined at these values.
Conclusion:
\[ x = \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z} \]Final Note: Solving trigonometric equations often involves the same strategies used in algebra—factoring, substitutions, and careful consideration of domain restrictions.