Looking at the image, we are given four angles (θ) and asked to find their reference angles (α), which is the acute angle between the terminal side of θ and the x-axis. The red handwritten work already shows the correct reference angles for each:
- For θ = 150° (Quadrant II): α = 30° → because 180° - 150° = 30°
- For θ = 315° (Quadrant IV): α = 45° → because 360° - 315° = 45°
- For θ = 4π/3 (Quadrant III): α = π/3 → because 4π/3 - π = π/3
- For θ = π/6 (Quadrant I): α = π/6 → because in Quadrant I, the reference angle is the same as the angle itself
The bottom part asks: “Determine how to find the reference angle in each quadrant.”
Let’s go quadrant by quadrant:
Quadrant I (0 to 90° or 0 to π/2):
The angle is already acute. So reference angle α = θ.
Quadrant II (90° to 180° or π/2 to π):
Reference angle α = 180° - θ (or π - θ in radians).
Quadrant III (180° to 270° or π to 3π/2):
Reference angle α = θ - 180° (or θ - π in radians).
Quadrant IV (270° to 360° or 3π/2 to 2π):
Reference angle α = 360° - θ (or 2π - θ in radians).
This matches what was drawn in the image — always measure the smallest angle to the x-axis.
Final Answer:
For Quadrant 1: α = θ
For Quadrant 2: α = 180° - θ (or π - θ)
For Quadrant 3: α = θ - 180° (or θ - π)
For Quadrant 4: α = 360° - θ (or 2π - θ)
Parent Tip: Review the logic above to help your child master the concept of coterminal angles worksheet.
