Angles and Unit Circle – Extra Practice

Step 1: Recall the coterminal angle formula:

Coterminal Angle = θ + 360°

Step 2: Plug in θ = −30°

−30° + 360°

Step 3: Simplify

= 330° ✅

Check: 330° is between 0° and 360° ✓ and lands on the same terminal side as −30° ✓

Step 1: Recall the coterminal angle formula:

Coterminal Angle = θ − 360°

Step 2: Plug in θ = 400°

400° − 360°

Step 3: Simplify

= 40°

Step 4: Need a negative coterminal — subtract 360° again

40° − 360° = −320° ✅

Check: −320° is negative ✓ and lands on the same terminal side as 400° ✓

Step 1: Recall the conversion formula:

Radians = Degrees × π/180

Step 2: Plug in 270°

270 × π/180

Step 3: Simplify — divide top and bottom by 90

= 3π/2

Answer: 3π/2 ✅

Step 1: Recall the conversion formula:

Degrees = Radians × 180/π

Step 2: Plug in 7π/6

7π/6 × 180/π

Step 3: Cancel π

= 7 × 180/6

Step 4: Simplify

= 7 × 30 = 210° ✅

Step 1: Identify the quadrant

150° is between 90° and 180° → Quadrant II

Step 2: Apply the Q II reference angle formula

Reference angle = 180° − θ

Step 3: Plug in

= 180° − 150° = 30° ✅

Step 1: Convert to degrees to identify the quadrant

5π/4 × 180/π = 225°

Step 2: 225° is between 180° and 270° → Quadrant III

Step 3: Apply the Q III reference angle formula

Reference angle = θ − π (in radians)

Step 4: Plug in

= 5π/4 − π = 5π/4 − 4π/4 = π/4 ✅

Step 1: Identify the quadrant

300° is between 270° and 360° → Quadrant IV

Step 2: Find the reference angle

360° − 300° = 60°

Step 3: Find cos(60°) from the unit circle

cos(60°) = 1/2

Step 4: Apply the sign — cos is positive in Q IV

cos(300°) = 1/2 ✅

Step 1: Identify the quadrant

5π/6 = 150° → Quadrant II

Step 2: Find the reference angle

π − 5π/6 = 6π/6 − 5π/6 = π/6

Step 3: Find sin(π/6) from the unit circle

sin(π/6) = 1/2

Step 4: Apply the sign — sin is positive in Q II

sin(5π/6) = 1/2 ✅

Step 1: Identify x and y from the point

x = −1/2  |  y = √3/2

Step 2: Apply all 6 definitions (r = 1 on unit circle)

sin θ = y = √3/2

cos θ = x = −1/2

tan θ = y/x = (√3/2)/(−1/2) = −√3

csc θ = 1/y = 1/(√3/2) = 2√3/3

sec θ = 1/x = 1/(−1/2) = −2

cot θ = x/y = (−1/2)/(√3/2) = −√3/3

Step 1: Identify the quadrant

7π/4 = 315° → Quadrant IV

Step 2: Find the reference angle

2π − 7π/4 = 8π/4 − 7π/4 = π/4

Step 3: Use unit circle values for π/4

sin(π/4) = √2/2  |  cos(π/4) = √2/2

Step 4: Apply Q IV signs (sin −, cos +)

sin(7π/4) = −√2/2  |  csc(7π/4) = −√2

cos(7π/4) = √2/2  |  sec(7π/4) = √2

tan(7π/4) = −1  |  cot(7π/4) = −1

Step 1: Recall the arc length formula

s = rθ

Step 2: Confirm θ is in radians ✓ (π/4 is already in radians)

Step 3: Plug in r = 8, θ = π/4

s = 8 × π/4

Step 4: Simplify

s = 2π ✅

Step 1: Recall the sector area formula

A = ½r²θ

Step 2: Confirm θ is in radians ✓ (2π/3 is already in radians)

Step 3: Plug in r = 9, θ = 2π/3

A = ½ × 81 × 2π/3

Step 4: Simplify

= ½ × 162π/3

= ½ × 54π

= 27π ✅