Trigonometry
Trigonometry connects angles, triangles, and periodic functions. Master right-triangle ratios (SOH-CAH-TOA), the unit circle, radian-degree conversion, the graphs of sine, cosine, and tangent with their amplitude, period, and phase shifts, and the fundamental identities. The law of sines and law of cosines extend this to non-right triangles.
Traps to watch
- Ambiguous case (SSA): the law of sines can yield zero, one, or two triangles-check whether the given angle is acute and compare side lengths.
- Radian mode: calculators and formulas assume radians unless stated; mixing modes corrupts answers.
- Reference angles and signs: sine is positive in quadrants I and II, cosine in I and IV; use reference angles to place the sign correctly.
- Period changes: in $y=\sin(bx)$, the period is $2\pi/|b|$, not $2\pi b$.
Worked example / rule of thumb
Find all solutions of $2\sin\theta = 1$ on $[0, 2\pi)$. $\sin\theta = 1/2$, so the reference angle is $\pi/6$. Sine is positive in quadrants I and II, giving $\theta = \pi/6$ and $\theta = 5\pi/6$. Rule of thumb: solve for the reference angle first, then use quadrant signs to generate all solutions within the interval. For non-right triangles, use the law of cosines when you know SAS or SSS, and the law of sines otherwise (watching for the ambiguous case). Convert degrees to radians by multiplying by $\pi/180$.