Radians are an alternative way to measure angles, based on the radius of a circle rather than degrees — a full circle is 2π radians (360°), so π radians equals 180°. The graphs of sin(x) and cos(x) are smooth, repeating waves oscillating between -1 and 1, with a period of 2π: sin(x) starts at 0, cos(x) starts at 1, and both repeat their pattern indefinitely, making them useful for modelling cyclical phenomena like tides or sound waves.
Example
The height of a swinging pendulum, or the depth of ocean tides throughout a day, can both be modelled using a sine or cosine function, since both rise and fall in a smooth, repeating pattern — exactly the shape these trigonometric graphs naturally produce.
Key terms
Radian:
A unit for measuring angles, based on a circle's radius; 2π radians = 360°.
Period:
The length of one complete cycle of a repeating function.
Amplitude:
The maximum distance a wave-like function moves from its centre line.
Questions
1. A full circle equals:
2π radians
π radians
4π radians
0 radians
2. π radians equals:
180°
360°
90°
45°
3. The graphs of sin(x) and cos(x) oscillate between:
-1 and 1
0 and 1
-2 and 2
0 and 360
4. The period of sin(x) and cos(x) is:
2π
π
1
360
5. sin(x) starts at:
0
1
-1
π
6. cos(x) starts at:
1
0
-1
π
7. Trigonometric functions like sin and cos are useful for modelling:
Cyclical phenomena, like tides
Only straight-line motion
Nothing that repeats over time
Only static, unchanging values
8. 90° in radians equals:
π/2
π
2π
π/4
9. 360° in radians equals:
2π
π
π/2
4π
10. Amplitude refers to:
The maximum distance a wave-like function moves from its centre line
The exact period of a function
The starting value of x only
A term unrelated to trigonometric graphs
11. 45° in radians equals:
π/4
π/2
π
2π
12. Why might radians be considered a more natural unit for angles in advanced mathematics than degrees?
Radians directly relate an angle to the radius and arc length of a circle, simplifying many calculus and trigonometric relationships
Degrees are always mathematically simpler to use in every context
Radians have no meaningful mathematical relationship to a circle's properties
There is no practical difference between using radians or degrees in any context
13. Why might ocean tide depth be modelled using a sine or cosine function rather than a straight line?
Tides rise and fall in a smooth, repeating pattern that matches the natural shape of these trigonometric graphs
A straight line always more accurately models cyclical, repeating phenomena
Tides change in a completely random, unpredictable pattern with no cyclical structure
Trigonometric functions have no practical application to real-world cyclical data
14. Why do sin(x) and cos(x) have the exact same shape, just shifted along the x-axis relative to each other?
Cos(x) is mathematically equivalent to a sine function shifted by π/2, reflecting their shared circular origin
Sin and cos are completely unrelated functions with no shared mathematical origin
The two functions always look completely different from one another with no relationship
This relationship only applies at a single specific angle, not generally
15. Why might a function like y = 3sin(x) have a different amplitude than y = sin(x), despite sharing the same period?
Multiplying the function by a constant stretches its vertical range without affecting how long one cycle takes
Amplitude and the multiplying constant have no mathematical relationship to each other
Multiplying by a constant always changes the period, never the amplitude
Every multiple of sin(x) always has an identical amplitude regardless of the constant
16. Why might sound engineers use trigonometric functions to model and analyse sound waves?
Sound waves are periodic vibrations, and sine/cosine functions naturally model this kind of repeating oscillation
Sound waves have no mathematical relationship to periodic, oscillating functions
Trigonometric functions can only model visual, not audio, phenomena
Sound engineering never involves any mathematical modelling of wave behaviour
17. Why might understanding the period of a trigonometric model be essential for predicting future values, like a future high tide?
Knowing how long one full cycle takes allows the pattern to be extended and predicted beyond the observed data
Period has no relevance to predicting future values in a cyclical model
Cyclical models can never be used to predict anything beyond the data already observed
The period of a function changes randomly and unpredictably over time
18. 270° in radians equals:
3π/2
π/2
2π
π/4
19. Why might a Ferris wheel's height above the ground over time be modelled using a cosine function shifted upward?
The height rises and falls in a smooth, repeating cycle, and the shift accounts for the wheel never reaching a height of zero or below
A Ferris wheel's motion is always best modelled with a straight line, not a trigonometric function
Cosine functions can never be shifted vertically to model realistic scenarios
A Ferris wheel's height changes in a completely random, non-repeating pattern
20. Why might a horizontal shift (phase shift) be added to a sine or cosine model when the cycle doesn't start at its usual reference point?
A phase shift adjusts where the cycle begins along the x-axis to match when the real-world pattern actually starts
Phase shifts have no practical use in modelling real-world periodic behaviour
Sine and cosine functions can never be shifted horizontally under any circumstance
The starting point of a cycle never needs to be adjusted to match real data
21. Why is it important to convert between degrees and radians correctly when using a calculator or software for trigonometric calculations?
Using the wrong angle mode produces a completely different, incorrect numerical result
Degrees and radians always produce identical numerical results regardless of mode
Calculators automatically detect and correct for whichever mode is intended
Angle mode has no effect on the outcome of a trigonometric calculation
Answer key (parent copy)
1. 2π radians
2. 180°
3. -1 and 1
4. 2π
5. 0
6. 1
7. Cyclical phenomena, like tides
8. π/2
9. 2π
10. The maximum distance a wave-like function moves from its centre line
11. π/4
12. Radians directly relate an angle to the radius and arc length of a circle, simplifying many calculus and trigonometric relationships
13. Tides rise and fall in a smooth, repeating pattern that matches the natural shape of these trigonometric graphs
14. Cos(x) is mathematically equivalent to a sine function shifted by π/2, reflecting their shared circular origin
15. Multiplying the function by a constant stretches its vertical range without affecting how long one cycle takes
16. Sound waves are periodic vibrations, and sine/cosine functions naturally model this kind of repeating oscillation
17. Knowing how long one full cycle takes allows the pattern to be extended and predicted beyond the observed data
18. 3π/2
19. The height rises and falls in a smooth, repeating cycle, and the shift accounts for the wheel never reaching a height of zero or below
20. A phase shift adjusts where the cycle begins along the x-axis to match when the real-world pattern actually starts
21. Using the wrong angle mode produces a completely different, incorrect numerical result