Trigonometry: Ratios, Identities, and Applications
Core concepts
Trigonometry on this exam spans right-triangle ratios, the unit circle, radian measure, graphing, identities, and solving triangles. Know that (\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}), (\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}), (\tan\theta=\frac{\text{opposite}}{\text{adjacent}}), and their reciprocals. Convert degrees to radians with (\theta_{rad}=\theta_{deg}\cdot\frac{\pi}{180}). Arc length is (s=r\theta) and sector area is (A=\frac{1}{2}r^2\theta) when (\theta) is in radians.
For non-right triangles, use the sine law (\frac{a}{\sin A}=\frac{b}{\sin B}) and the cosine law (c^2=a^2+b^2-2ab\cos C). The cosine law handles SAS and SSS; the sine law handles ASA, AAS, and SSA. The SSA case is the ambiguous case and may yield zero, one, or two triangles. Be able to graph (y=a\sin(b(x-c))+d), identifying amplitude (|a|), period (\frac{2\pi}{b}), phase shift, and midline (d).
Traps to watch
- Using degree mode when the problem is in radians.
- Forgetting the ambiguous case when given SSA.
- Applying the sine law when the given information requires the cosine law.
- Mixing amplitude with vertical shift.
Rule of thumb
Label the triangle first: if you know a matching side-angle pair, use the sine law; if you know two sides and the included angle or all three sides, use the cosine law.
Worked example
A sector has radius 6 cm and central angle (\frac{\pi}{3}). Arc length (s=6\cdot\frac{\pi}{3}=2\pi\approx 6.28) cm; area (=\frac{1}{2}(36)\frac{\pi}{3}=6\pi\approx 18.85) cm².