Chapter 7: The Unit Circle and Radian Measure

Question 1

Convert from radian to degrees and degrees to radians

Answer 1

π radian = 180°

Question 2

State the formulas for arc length and sector area in both radian and degrees

Answer 2

Arc length:
Degree: θ/360 x 2πr
Radian: θr

Sector area:
Degree: θ/360 x πr²
Radian: (1/2)(θr²)

Question 3

State the properties of the unit circle

Answer 3

• Centre is at origin (0,0)
• P: (cosθ, sinθ)
• Equation is x²+y² = 1
  (Therefore, cos²θ+sin²θ = 1)
• Radius is 1 unit
• Domain: -1≤ x≤ 1 (-1≤ cosθ≤ 1); Range: -1≤ y≤ 1 (-1≤ sinθ≤ 1)

Question 4

Define tangent in terms of the unit circle

Answer 4

Since:
sinθ/cosθ = tanθ
P = (cosθ , sinθ)
tanθ = y/x = gradient function of the unit circle

Question 5

Identify the 4 quadrants of the unit circle and state which trigonometric ratios are positive in each one

Answer 5

Quadrant 1:
0°< θ < 90° OR 0 < θ < π/2
[All trigonometric ratios are positive)

Quadrant 2:
90°< θ < 180° OR π/2 < θ < π
[sinθ is positive, as: sin(180°- θ) = sinθ]
[cosθ is negative, as cos(180°- θ) = -cosθ

Quadrant 3:
180°< θ < 270° OR π < θ < 3π/2
[-tanθ is positive, as sinθ/-cosθ = tanθ]

Quadrant 4:
270°< θ < 360° OR 3π/2 < θ < 2π
[cosθ is positive]

ASTC
(adding 2π is 1 full revolution –> same point on the unit circle)

Question 6

State the special angles

Answer 6

sin(0)=0
sin(π/6)=1/2
sin(π/4)=√2/2
sin(π/3)=√3/2
sin(π/2)=1

cos(0)=1
cos(π/6)=√3/2
cos(π/4)=√2/2
cos(π/3)=1/2
cos(π/2)=0

tan(0)=0
tan(π/6)=√3/3
tan(π/4)=1
tan(π/3)=√3
tan(π/2)=undefined

Question 7

Apply the pythagorean identity to find other trigonometric ratios when given another trigonometric ratio

Answer 7

sin²θ+cos²θ = 1

Eg.
If sinθ=2/3
(2/3)² +cos²θ = 1
cos²θ = 1 - 4/9 = 5/9
cosθ = √5/9