M151Q Precalculus · Unit 4: Trigonometric functions · on Final Exam
TRIG 4.1–4.3 Angles, rotation, and the graphs of trigonometric functions
Study notes for Angles, rotation, and the graphs of trigonometric functions in M151Q Precalculus at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- Angles in standard position open counterclockwise from the positive $x$-axis; negative angles go clockwise; coterminal angles differ by $360^\circ$. The reference angle is the acute angle to the $x$-axis; trig values of any angle equal those of its reference angle up to sign, and the quadrant fixes the sign (All Students Take Calculus: I all positive, II sine, III tangent, IV cosine).
- On the unit circle $(\cos\theta, \sin\theta)$ is the point at angle $\theta$. So $\sin$ and $\cos$ are periodic with period $360^\circ$, bounded between $-1$ and $1$.
- Graphs: $y = \sin\theta$ starts at 0 rising; $y = \cos\theta$ starts at 1; both have amplitude 1 and period $360^\circ$. $y = \tan\theta$ has period $180^\circ$ with vertical asymptotes at $\pm90^\circ$.
- Use the graphs to solve $\sin\theta = k$ over a range: two solutions per period for $|k| < 1$, found from the reference angle.
Key formulas
Reference angles
$$\text{QII: } 180^\circ - \theta,\quad \text{QIII: } \theta - 180^\circ,\quad \text{QIV: } 360^\circ - \theta$$
Unit circle
$$P(\theta) = (\cos\theta,\ \sin\theta),\qquad \sin^2\theta + \cos^2\theta = 1$$
Solving
$$\sin\theta = k \Rightarrow \theta = \sin^{-1}k\ \text{or}\ 180^\circ - \sin^{-1}k\ (+360^\circ n)$$
Worked example
Find all $\theta$ in $[0^\circ, 360^\circ)$ with $\cos\theta = -\tfrac12$, and evaluate $\sin 210^\circ$ exactly.
Show the solution
Reference angle $60^\circ$; cosine negative in QII and QIII: $\theta = 120^\circ, 240^\circ$. $\sin210^\circ$: QIII, reference $30^\circ$, sine negative: $-\tfrac12$.
Common mistakes
- Reporting only the calculator’s answer (one solution) when the range has two.
- Wrong sign for the quadrant.
- Confusing period (horizontal) with amplitude (vertical).
Exam tipDraw the angle. A five-second sketch of the quadrant beats memorizing sign rules.
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