Trigonometry
Trigonometry
Trigonometry in Pre-Calculus 40S spans the unit circle, radian measure, graphing sinusoidal functions, identities, and solving trigonometric equations.
Core concepts. Know the unit circle exactly for multiples of $30^\circ$ and $45^\circ$ (equivalently $\pi/6$ and $\pi/4$), including signs by quadrant. Convert with $180^\circ=\pi$ radians. Arc length is $s=r\theta$ and sector area is $A=\tfrac12 r^2\theta$, with $\theta$ in radians. For $y=a\sin b(x-c)+d$, the amplitude is $|a|$, the period is $\tfrac{2\pi}{|b|}$ (or $\tfrac{360^\circ}{|b|}$), the phase shift is $c$, and the vertical displacement is $d$. The set of basic identities includes the Pythagorean identities, reciprocal identities, and quotient identities; the double-angle and sum identities extend these.
Traps. Mixing degree and radian mode is the most common error in calculator work. A negative $a$ reflects the graph and is not part of the amplitude, since amplitude is always non-negative. When solving $\sin\theta=k$, you must supply all solutions in the requested interval, using reference angles and symmetry; forgetting the second solution in the interval is a classic loss of marks. Identities are proved by working on one side only, not by moving terms across the equals sign.
Rule of thumb. For any trig equation, find the reference angle first, then use the sign of the function to place solutions in the correct quadrants, then add or subtract the period until you exhaust the interval.
Worked example. Solve $2\sin\theta-1=0$ for $0\le\theta<2\pi$. Then $\sin\theta=\tfrac12$, reference angle $\tfrac{\pi}{6}$, positive in quadrants I and II, so $\theta=\tfrac{\pi}{6}$ or $\theta=\tfrac{5\pi}{6}$.