Radians & the Unit Circle — Practice
Measure angles by the arc they cut, and trigonometry escapes the right triangle.
What these questions 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
Convert 180° to radians. Give the coefficient of π.
Worked answer
1
180° = π radians
Problem 2
Convert 90° to radians. Give your answer in the form a/b times pi, as the fraction only.
Worked answer
1/2
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Problem 3
Convert π4 radians to degrees.
Worked answer
45
1804=45
Problem 4
Convert 2π3 radians to degrees.
Worked answer
120
2 × 1803=120
Problem 5
How many radians are there in a full turn? Give the coefficient of π.
Worked answer
2
2π
Problem 6
Convert 270° to radians. Give the answer as the fraction multiplying π.
Worked answer
3/2
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Problem 7
Find the reference angle for 150°.
Worked answer
30
180-150=30
Problem 8
Find the reference angle for 210°.
Worked answer
30
210-180=30
Problem 9
Find the reference angle for 300°.
Worked answer
60
360-300=60
Problem 10
In quadrant III, which is true?
- Both sine and cosine are negative
- Sine positive, cosine negative
- Both positive
- Sine negative, cosine positive
Worked answer
Both sine and cosine are negative
Quadrant III has x<0 and y<0
cos follows x, sin follows y
Problem 11
Find sin 90°.
Worked answer
1
The point is (0,1), and sin is the y-coordinate
Problem 12
Find cos 180°.
Worked answer
-1
The point is (-1,0)
cos is the x-coordinate
Problem 13
A circle of radius 6 has a central angle of π3. Find the arc length. Give the coefficient of π.
Worked answer
2
6 · π3=2π
Problem 14
A circle of radius 10 has a central angle of 1.5 radians. Find the arc length.
Worked answer
15
10 × 1.5 = 15
Problem 15
Find the angle between 0° and 360° that is coterminal with 430°.
Worked answer
70
430-360=70
Problem 16
Find the angle between 0° and 360° that is coterminal with -50°.
Worked answer
310
-50+360=310