Trigonometric functions become much easier when students stop treating sine, cosine and tangent as isolated calculator buttons. In Additional Mathematics, the important ideas are function behaviour, exact values, graph shape, period, amplitude, transformation and equation structure.
This guide builds the graph-and-function foundation that supports trigonometric identities, trigonometric equations and R-form. It belongs to the wider Additional Mathematics Hub.
1. Think of sine, cosine and tangent as functions
A function takes an input angle and produces an output. For y = sin x, each value of x is mapped to a sine value. The same idea applies to cosine and tangent. Once students see trigonometry as a family of functions, graph questions and equation questions become connected rather than separate chapters.
2. Exact values are structural anchors
Students should know the standard exact values associated with familiar angles. These values help with graphs, identities and equations because they provide reliable reference points without calculator dependence.
| Angle | sin | cos | tan |
|---|---|---|---|
| 0 | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
The important goal is not only memorisation. Students should know how these values fit into the wider periodic behaviour of the functions.
3. The sine graph
The graph of y = sin x oscillates between -1 and 1. Its amplitude is 1. In degrees, one full cycle takes 360°. In radians, one full cycle takes 2π.
Key values over one cycle are:
- sin 0 = 0
- sin 90° = 1
- sin 180° = 0
- sin 270° = -1
- sin 360° = 0
The graph then repeats.
4. The cosine graph
The graph of y = cos x has the same amplitude and period as sine, but it begins at 1 when x = 0.
- cos 0 = 1
- cos 90° = 0
- cos 180° = -1
- cos 270° = 0
- cos 360° = 1
This difference in starting position is one reason students should understand graph shape instead of trying to memorise disconnected coordinate lists.
5. The tangent graph
Tangent behaves differently. Its basic period is 180° or π radians, and the graph has vertical asymptotes where cosine is zero. This is consistent with tan x = sin x / cos x.
When cos x = 0, tangent is undefined. That relationship explains the asymptotes rather than treating them as arbitrary graph features.
6. Amplitude and period
For a function such as y = a sin bx, the value of |a| controls the amplitude. The value of b changes the period.
In degrees, the period of y = sin bx or y = cos bx is:
360° / |b|
In radians, it is:
2π / |b|
For tangent, the basic period is halved:
180° / |b| or π / |b|.
7. Worked example: y = 3 sin 2x
The amplitude is 3. In degrees, the period is 360°/2 = 180°.
This means the graph completes two full sine cycles between 0° and 360°, while its maximum and minimum values are 3 and -3.
8. Vertical and horizontal shifts
For y = a sin b(x – h) + k:
- |a| controls amplitude.
- b controls period.
- h moves the graph horizontally.
- k moves the graph vertically.
A useful strategy is to identify these four jobs separately rather than trying to sketch the whole transformed graph in one mental jump.
9. Worked example: y = 2 cos(x – 30°) + 1
The amplitude is 2. The period remains 360°. The graph is shifted 30° to the right and 1 unit upward.
Because the midline is y = 1, the maximum value is 3 and the minimum value is -1.
10. Radians and graph control
Students should be comfortable moving between degrees and radians. The key reference is:
180° = π radians.
Therefore 90° = π/2, 60° = π/3 and 45° = π/4. Calculator mode must match the angle unit used by the question. A correct method with the wrong calculator mode can produce a completely wrong numerical answer.
11. Graphs help solve equations
Consider sin x = 1/2 over a specified interval. The exact value tells us the reference angle is 30°, while the sine graph tells us where positive sine values occur over a full cycle. The graph view and the symbolic view support each other.
This is why graph fluency should be built before students rely heavily on equation-solving routines.
12. Common mistakes
| Mistake | Cause | Repair |
|---|---|---|
| Using 360°/b for tangent | Applying sine/cosine period automatically | Remember tangent basic period is 180° |
| Confusing amplitude with vertical shift | Reading coefficients without roles | Separate a, b, h and k |
| Wrong calculator mode | Degrees/radians not checked | Write the unit before calculation |
| Missing tangent asymptotes | Graph memorised without tan = sin/cos | Locate where cos x = 0 |
| Only one solution to a trig equation | Reference angle used without graph cycle | Use graph or quadrant structure across the interval |
13. Practice set
- State the amplitude and period of y = 4 sin 3x in degrees.
- State the amplitude and period of y = -2 cos(x/2) in degrees.
- State the period of y = tan 4x in degrees.
- For y = 3 sin(x – 20°) + 2, state the horizontal shift, vertical shift, maximum and minimum values.
- Convert 150° to radians.
14. Answers
- Amplitude 4; period 120°.
- Amplitude 2; period 720°.
- 45°.
- 20° right; 2 up; maximum 5; minimum -1.
- 5π/6.
15. What mastery looks like
A student has strong trigonometric function control when they can predict graph shape before using a calculator, connect exact values to graph positions, identify amplitude and period reliably, understand transformations, and use the graph to support equation solving.
Continue through the Geometry and Trigonometry Guide, Trigonometric Identities, Trigonometric Equations, or return to the A-Math Hub.