Trigonometry / Unit 7: Trigonometric Equations
7.5 Equations that Need Identities
Notes
If an equation mixes different functions or angles, use an identity to rewrite it in one function.
- Replace $\sin^2 x$ with $1 - \cos^2 x$ (or the reverse) to get a quadratic in one function.
- Replace $\sin 2x$ with $2\sin x\cos x$, then factor.
- Example: $\sin 2x = \cos x \Rightarrow 2\sin x\cos x - \cos x = 0 \Rightarrow \cos x (2\sin x - 1) = 0$.
Practice
Solve $2\sin^2 x + 3\cos x - 3 = 0$ on $[0, 2\pi)$.
Replace $\sin^2 x$ with $1 - \cos^2 x$: $2\cos^2 x - 3\cos x + 1 = 0$, so $(2\cos x - 1)(\cos x - 1) = 0$.
Solve $\sin 2x = \sin x$ on $[0, 2\pi)$.
$2\sin x\cos x - \sin x = 0$, so $\sin x (2\cos x - 1) = 0$. Then $\sin x = 0$ or $\displaystyle \cos x = \frac{1}{2}$.
Why is it a mistake to divide both sides of $\sin x\cos x = \sin x$ by $\sin x$?
Factor instead: $\sin x(\cos x - 1) = 0$ keeps the solutions from $\sin x = 0$.
Ready for more? Take the Unit 7 test.