Radians & the Unit Circle — Challenge
Measure angles by the arc they cut, and trigonometry escapes the right triangle.
What these problems cover
These questions use 13 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Coefficient — The number multiplying a variable — the 3 in 3n.
- π — The circumference of any circle divided by its diameter — about 3.14159.
- Radius — Centre to edge. Half the diameter.
- Arc — Part of the edge; a "pizza slice" bounded by two radii.
- Sine — sin θ = oppositehypotenuse.
- Cosine — cos θ = adjacenthypotenuse.
- Central angle — An angle whose vertex is the centre. Equal to its arc.
- Radian — The angle whose arc length equals the radius. A full turn is 2π radians.
- Unit circle — The circle of radius 1 centred at the origin.
- Reference angle — The acute angle between the terminal side and the x-axis.
- x-coordinate — The first number — how far across.
- y-coordinate — The second number — how far up or down.
- Quadrant — One of the four regions the axes divide the plane into.
Problem 1
Which point on the unit circle corresponds to an angle of 90 degrees?
- (1,0)
- (0,1)
- (-1,0)
- (0,-1)
- (1,1)
Worked answer
(0,1)
Angles on the unit circle are measured anticlockwise from the point (1,0).
A quarter turn carries that point to the top of the circle.
The top is (0,1). This matches cos 90° = 0 and sin 90° = 1.
Problem 2
How many degrees is 5π6 radians?
- 120°
- 135°
- 150°
- 210°
- 300°
Worked answer
150°
Multiply by 180π: 5π6 · 180π = 5 × 1806. = 5 × 30 = 150. Sensible check: 56 of a half turn is 56 of 180°.
Problem 3
A circle of radius 12 has an arc of length 8. What is the central angle, in radians?
- 13
- 23
- 32
- 43
- 96
Worked answer
23
Arc length in radians is s=rθ, so θ=sr. θ=812=23. No π appears because a radian is a ratio, not a fraction of a turn.
Problem 4
Which angle between 0° and 360° is coterminal with -100°?
- 80°
- 160°
- 200°
- 260°
- 280°
Worked answer
260°
Coterminal angles differ by whole turns, so add 360°. -100+360=260. A check: -100° is 100° clockwise from the positive x-axis, which lands in the third quadrant — and so does 260°.
Problem 5
In which quadrant does an angle of 4 radians lie?
- I
- II
- III
- IV
- On an axis
Worked answer
III
The quadrant boundaries in radians are π2 ≈ 1.57, π ≈ 3.14 and 3π2 ≈ 4.71. Since 3.14 < 4 < 4.71, the angle is past the negative x-axis but not yet at the bottom. That is quadrant III.
Problem 6
If cosθ=-12 and sinθ>0, what is θ for 0° ≤ θ < 360°?
- 60°
- 120°
- 150°
- 210°
- 240°
Worked answer
120°
The reference angle satisfies cos = 12, which is 60°. Cosine negative and sine positive puts the angle in quadrant II. In quadrant II the angle is 180-60=120°.
Problem 7
The minute hand of a clock is 10 cm long. How far does its tip travel in 20 minutes?
- 10π3 cm
- 20π3 cm
- 20π cm
- 100π3 cm
- 5π cm
Worked answer
20π3 cm
In 20 minutes the hand sweeps 2060=13 of a full turn, which is 2π3 radians. Arc length is s=rθ=10 · 2π3. =20π3 cm, about 20.9 cm.
Problem 8
What is sin7π6?
- 12
- -12
- √32
- -√32
- -1
Worked answer
-12
7π6 is π6 past π, so the reference angle is π6 and the angle sits in quadrant III. The size is sinπ6=12. In quadrant III the y-coordinate is negative, so the answer is -12.