Trigonometric Functions: Identities, Equations and Graphs
Core concepts
You need exact values, the ASTC sign pattern, radian measure, arc length $l=r\theta$ and sector area $A=\tfrac12 r^2\theta$. Know the Pythagorean identities, the reciprocal and quotient identities, and the compound-angle and double-angle formulas. Be able to prove identities, solve equations over a stated interval, and sketch $y=a\sin(b(x-c))+d$ with amplitude $|a|$, period $\frac{2\pi}{b}$ and phase shift $c$.
Traps
- Losing solutions: a general solution like $\theta=\alpha+2k\pi$ may also need $\theta=\pi-\alpha+2k\pi$; always find all solutions in the given interval.
- Squaring both sides can introduce extraneous roots - check each answer in the original equation.
- Working only on one side of an identity is safer; never multiply both sides by an expression that could be zero.
- Radian mode: derivatives and integrals of trig functions require radians.
- $\sin^{-1}$ denotes inverse sine, not $1/\sin$.
Rule of thumb
For $a\sin\theta+b\cos\theta$, write it as $R\sin(\theta+\phi)$ with $R=\sqrt{a^2+b^2}$ and $\tan\phi=b/a$; this converts the equation to a single sine and reveals maximum $R$.
Worked example
Solve $2\sin^2x-\sin x-1=0$ for $0\le x\le 2\pi$. Factor: $(2\sin x+1)(\sin x-1)=0$. So $\sin x=-\tfrac12$ giving $x=\frac{7\pi}{6},\frac{11\pi}{6}$, or $\sin x=1$ giving $x=\frac{\pi}{2}$. All four lie in the interval, so the solution set is ${\frac{\pi}{2},\frac{7\pi}{6},\frac{11\pi}{6}}$.