Trigonometric Functions
Graphs of the Trigonometric Functions
The sine and cosine waves, the tangent curve with its asymptotes, and what the shapes reveal about domain, range and symmetry.
What this page covers
- Sketch y = sin x and y = cos x over one period
- Sketch y = tan x and locate its asymptotes
- Read domain, range and period from a graph
- Identify each function as even or odd from its graph
Sine
As the point travels round the unit circle, its height y = sin t rises from 0 to 1, falls back through 0 to -1, and returns. Plotting height against distance travelled produces the sine wave.
| t | 0 | π6 | π4 | π3 | π2 | π | 3pi2 | 2pi |
|---|---|---|---|---|---|---|---|---|
| sin t | 0 | 12 | 1√2 | √32 | 1 | 0 | −1 | 0 |
- Domain
- All real x
- Range
- [-1, 1]
- Period
- 2pi
- Zeros
- x = 0, π, 2pi, …, that is x = npi
- Maximum
- 1, at x = π2 + 2npi
- Minimum
- -1, at x = 3pi2 + 2npi
- Symmetry
- About the origin — sine is odd
The source marks the period on its sketch as 2pi and labels the amplitude separately. A useful sketching habit is to divide each period into four: zero, maximum, zero, minimum, zero. Those five points determine the shape.
Cosine
The same wave, started a quarter period earlier. At t = 0 the point is at (1, 0), so cosine begins at its maximum rather than at zero.
| t | 0 | π4 | π2 | π | 3pi2 | 2pi |
|---|---|---|---|---|---|---|
| cos t | 1 | 1√2 | 0 | −1 | 0 | 1 |
- Domain
- All real x
- Range
- [-1, 1]
- Period
- 2pi
- Zeros
- x = π2 + npi
- Maximum
- 1, at x = 2npi
- Symmetry
- About the y-axis — cosine is even
This relationship is why the addition formulae for sine and cosine are so closely linked, and why sin(π2 - θ) = cos θ — an identity the source derives directly from the addition formula in Week 9.
Tangent
Tangent behaves quite differently, because it is a quotient whose denominator vanishes periodically.
- Domain
- x ≠ π2 + npi
- Range
- All real values
- Period
- π — half that of sine and cosine
- Zeros
- x = npi, where sin x = 0
- Asymptotes
- x = π2 + npi, where cos x = 0
- Symmetry
- About the origin — tangent is odd
The typed notes reason it out in exactly this order: find where cos x = 0 to locate the asymptotes, then where sin x = 0 to locate the zeros. Between consecutive asymptotes the curve rises from -∞ to +∞, passing through zero at the midpoint.
A half-turn maps (x, y) to (-x, -y), and the ratio -y-x = yx is unchanged. So the tangent takes the same value at t and t + π, giving a period of π.
The source's sketch shows repeated branches separated by dashed vertical lines at π2, 3pi2 and so on. Each branch is a complete copy of the one before, shifted by π.
The three compared
| Feature | sin x | cos x | tan x |
|---|---|---|---|
| Period | 2pi | 2pi | π |
| Range | [-1, 1] | [-1, 1] | All real |
| Zeros | npi | π2 + npi | npi |
| Asymptotes | None | None | π2 + npi |
| Parity | Odd | Even | Odd |
| Value at 0 | 0 | 1 | 0 |
| Bounded? | Yes | Yes | No |
The parity row is worth connecting to the circle. Reflecting a point in the x-axis sends t to -t, keeping x and negating y. So cos(-t) = cos t and sin(-t) = -sin t, and tangent, being their ratio, is odd. The source states both directly in Week 8.
Sketching from five points
- Mark the period on the horizontal axis and divide it into four equal parts.
- Plot the five key points. For sine: zero, maximum, zero, minimum, zero. For cosine: maximum, zero, minimum, zero, maximum.
- Draw a smooth curve through them — never straight segments.
- Repeat the shape left and right, since the function is periodic.
For tangent the procedure differs: mark the asymptotes first, then the zero midway between each pair, then sketch a branch rising steeply from one asymptote to the next.
A sine curve is not a series of arcs and not a zigzag. It has zero gradient exactly at its peaks and troughs and its steepest gradient at the zeros, which is what makes it smooth rather than pointed.
Common mistakes
| Mistake | Correct | Check |
|---|---|---|
| Starting the cosine curve at zero | It starts at 1 | cos 0 = 1 |
| Giving tangent period 2pi | It is π | A half-turn leaves yx unchanged |
| Bounding tangent between -1 and 1 | It is unbounded | Its range is all real values |
| Drawing tangent through its asymptotes | It is undefined there | The denominator is zero |
| Calling sine even | Sine is odd | sin(-t) = -sin t |
| Sketching with straight segments | The curve is smooth | Zero gradient at the turning points |
Frequently asked questions
How do sine and cosine graphs differ?
Only by a shift. cos x = sin(x + π2), so the cosine curve is the sine curve moved a quarter period to the left. Their shapes are identical.
Why does tangent have vertical asymptotes?
Because tan x = sin xcos x and the denominator vanishes at π2 and every angle π from it. Near those points the quotient grows without bound.
Why is tangent's period π and not 2pi?
A half-turn negates both coordinates, and the ratio yx is unchanged by negating both. So the tangent repeats after π.
Which functions are even and which are odd?
Cosine is even: cos(-t) = cos t. Sine and tangent are odd. The graphs show it directly — cosine is symmetric about the y-axis, the other two about the origin.
Source. Handwritten teaching notes, Week 7 pages 3, 5 and 9, with the typed supplementary notes on the basics of trigonometry.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.