Study Guide
Learning Objectives
- Find $P=(r\cos\theta,\, r\sin\theta)$ on a circle centered at the origin (3.3.A.1).
- Compute exact $\sin$ and $\cos$ for multiples of $\pi/6$ and $\pi/4$ using special triangles and quadrant signs (3.3.A.2).
- Apply results in geometric or applied contexts (skill 3.B).
Suggested Practice Sequence
1. Unit circle QI: $\sin$ and $\cos$ of $\pi/6$, $\pi/4$, $\pi/3$. 2. Axis angles: $0$, $\pi/2$, $\pi$, $3\pi/2$. 3. Other quadrants: same absolute values, correct signs. 4. Scale by $r$: write full coordinates. 5. Coterminal / negative angles reduced to a known position. 6. Short applied: "point on a wheel of radius $r$."
Classroom / Check-In Moves
- Special-triangle flash: legs vs hypotenuse ratios.
- Sign-only cards by quadrant.
- "$r$-scale" exit ticket: give $\theta$ and $r$, ask for $P$.
Misconception Watch-List
- Using unit-circle values when $r\neq 1$ without multiplying.
- Wrong reference angle (e.g., treating $2\pi/3$ like $\pi/3$ without QII signs).
- Swapping $\sin$ and $\cos$ for $\pi/6$ vs $\pi/3$.
- Leaving radicals in the denominator incorrectly (or inconsistently with class norms).
Vocabulary to Own
- coordinates on a circle; radius $r$
- exact value
- reference angle / special triangles (isosceles right; equilateral half)
- quadrant signs
- multiples of $\pi/6$ and $\pi/4$
Materials Map
- Practice: exact values and $(r\cos, r\sin)$
- Word: radius contexts
- Mixed / multi-*: link to 3.2 definitions
- Review & practice test: fluency chart
- Interactive: exact-value checks