1. Understanding Trigonometric Graphs
Graphs of trigonometric functions help us visualise how the values of sin, cos and tan change as the angle increases. These graphs repeat after certain intervals (periods) and show waves, peaks and troughs.
While we do not draw the actual curves here, each graph will be described clearly so you can imagine the shape and behaviour.
2. Graph of y = sin x
The sine graph is a smooth wave-like curve that starts from 0 and oscillates between –1 and +1.
- Starts at: (0, 0)
- Maximum: 1 at \(x = \dfrac{\pi}{2}\)
- Minimum: –1 at \(x = \dfrac{3\pi}{2}\)
- Period: \(2\pi\)
- Range: [–1, 1]
2.1. Key Features of the Sine Wave
- The graph crosses the x-axis at multiples of \(\pi\): 0, \(\pi\), \(2\pi\), ...
- Repeats its pattern every \(2\pi\).
- Symmetrical about the origin (it is an odd function).
3. Graph of y = cos x
The cosine graph looks similar to the sine graph but starts at its maximum value.
- Starts at: (0, 1)
- Maximum: 1 at x = 0, \(2\pi\), ...
- Minimum: –1 at \(x = \pi\)
- Period: \(2\pi\)
- Range: [–1, 1]
3.1. Key Features of the Cosine Curve
- Crosses the x-axis at \(\dfrac{\pi}{2}\), \(\dfrac{3\pi}{2}\), ...
- Repeats after \(2\pi\).
- Symmetrical about the y-axis (it is an even function).
4. Graph of y = tan x
The tangent graph looks very different from sine and cosine. It has vertical asymptotes (lines the curve approaches but never touches) and repeats more frequently.
- Starts repeating every: \(\pi\)
- Asymptotes at: \(x = \dfrac{\pi}{2} + n\pi\)
- No maximum or minimum: the graph goes to ±∞
- Range: all real numbers
4.1. Behaviour of the Tangent Graph
- Rises sharply from –∞ to +∞ within each interval.
- Crosses the x-axis at multiples of \(\pi\).
- Undefined at \(\dfrac{\pi}{2}, \dfrac{3\pi}{2}, ...\)
5. Comparing sin x, cos x and tan x
Here is a summary comparison to help remember:
| Function | Starts at | Period | Range | Special Features |
|---|---|---|---|---|
| sin x | 0 | \(2\pi\) | [–1, 1] | Odd function, wave passing through origin |
| cos x | 1 | \(2\pi\) | [–1, 1] | Even function, wave starting at peak |
| tan x | 0 | \(\pi\) | All real numbers | Vertical asymptotes, crosses x-axis every \(\pi\) |
6. Understanding Amplitude and Period
Amplitude is half the height from maximum to minimum. It applies to sine and cosine:
\(\text{Amplitude} = 1 \text{ (for sin and cos)}\)
Period tells how long it takes for a graph to repeat:
- sin and cos → \(2\pi\)
- tan → \(\pi\)
7. Example: Visualising sin x and cos x Together
Imagine plotting both on the same graph:
- cos x starts at 1
- sin x starts at 0
- Both reach 1 and –1 at different points
- cos graph appears like the sin graph shifted left by \(\dfrac{\pi}{2}\)
\( \cos x = \sin(x + \dfrac{\pi}{2}) \)