Graphs of Trigonometric Functions
Theory
The graphs of \(y = \sin x\) and \(y = \cos x\) are waves between \(-1\) and \(1\) that repeat every \(360^\circ\); \(y = \tan x\) repeats every \(180^\circ\) and has asymptotes. This Year 10A maths page covers their key features, amplitude and period, transformations such as \(y = a\sin x\), \(y = \sin(bx)\) and \(y = \cos x + c\), reading equations and inequalities from a graph, and simple models such as tides and temperatures.
Plotting \(\sin x\) for every angle \(x\) gives a wave. It starts at \(0\), reaches a maximum of \(1\) at \(90^\circ\), returns to \(0\) at \(180^\circ\), has a minimum of \(-1\) at \(270^\circ\) and is back at \(0\) at \(360^\circ\). Then it repeats.
\(y = \cos x\) has the same shape, starting at its maximum: \(1\) at \(0^\circ\), \(0\) at \(90^\circ\), \(-1\) at \(180^\circ\), \(0\) at \(270^\circ\), \(1\) at \(360^\circ\). Both have range \(-1 \le y \le 1\).
\(y = \tan x\) is \(0\) at \(0^\circ\), \(180^\circ\) and \(360^\circ\). Where \(\cos x = 0\) it is undefined, and the graph has a vertical asymptote: a dashed line the curve approaches but never meets. Its range is all real numbers.
The period is the length of one complete wave along the \(x\)-axis: \(360^\circ\) for sine and cosine, \(180^\circ\) for tangent. The amplitude is the height from the centre line of a wave to a maximum, half the distance from the minimum to the maximum. For \(y = \sin x\) and \(y = \cos x\) the amplitude is \(1\); tangent has none.
The graphs show the angle facts as shapes. The sine graph is symmetric about \(x = 90^\circ\), so \(\sin(180^\circ - x) = \sin x\). The cosine graph is the sine graph moved \(90^\circ\) left. Turning \(\cos x\) upside down gives \(\cos(180^\circ + x) = -\cos x\), and \(\tan(-x) = -\tan x\) reflects the tangent graph.
Transformations, shown for sine (cosine and tangent work the same way):
| Graph | Effect on \(y = \sin x\) |
|---|---|
| \(y = a\sin x\) | vertical dilation by factor \(a\): range \(-a \le y \le a\) (\(a \gt 0\)) |
| \(y = \sin x + c\) | translation \(c\) units up (down if \(c < 0\)) |
| \(y = \sin(x - h)\) | translation \(h\) to the right; \(\sin(x + h)\) moves left |
| \(y = \sin(bx)\) | horizontal dilation by factor \(\dfrac{1}{b}\): one wave every \(\dfrac{360^\circ}{b}\) |
| \(y = -\sin x\) | reflection in the \(x\)-axis |
How to sketch a transformed trig graph
- Start with the five key points of the basic graph: \(0^\circ\), \(90^\circ\), \(180^\circ\), \(270^\circ\), \(360^\circ\).
- Apply each change: multiply \(y\) by \(a\), add \(c\) to \(y\), move \(x\) by \(h\), or divide \(x\) by \(b\).
- Join the points with a smooth wave, add asymptotes for tangent, and check the range and intercepts.
Reading information from a graph
- Maximum and minimum: read the highest and lowest \(y\)-values and where they occur.
- \(x\)-intercepts: set \(y = 0\) and solve, or read where the curve crosses the axis.
- Which graph is it? Make a table of values at \(0^\circ\), \(90^\circ\), \(180^\circ\), \(270^\circ\), \(360^\circ\) (or every \(45^\circ\) for \(\sin 2x\)) and match it.
- Finding \(a\) and \(b\) in \(y = a\sin(bx)\): \(a\) is the maximum value; \(b\) is the number of full waves in \(360^\circ\).
- Where two graphs meet: solve the equation, e.g. \(\sin x = \cos x\) gives \(\tan x = 1\).
- Inequalities: solve the equation for the boundary, then read from the graph where the curve is above or below the line (Example 3).
- Models: the largest and smallest values come from \(\sin\) or \(\cos\) being \(1\) or \(-1\); solve for the time (Example 4).
(i) describe the graph as transformations of \(y = \cos x\)
(ii) find the range
(iii) find the \(x\)-intercepts, correct to one decimal place.
(i) Multiply every \(y\)-value by \(3\) (a vertical dilation by factor \(3\)), then add \(1\) (a translation \(1\) unit up).
(ii) Start from \(-1 \le \cos x \le 1\) and do the same to each part:
| \(-3 \le 3\cos x\) | \(\le\) | \(3\) |
| \(-2 \le 3\cos x + 1\) | \(\le\) | \(4\) |
The range is \(\mathbf{-2 \le y \le 4}\).
(iii) Let \(y = 0\). Cosine is negative in quadrants 2 and 3.
| \(\cos x\) | \(=\) | \(-\dfrac{1}{3}\) |
| \(\alpha\) | \(=\) | \(\cos^{-1}\left(\dfrac{1}{3}\right) = 70.528\ldots^\circ\) |
| \(x\) | \(=\) | \(180^\circ - \alpha \approx \mathbf{109.5^\circ}\) |
| \(\text{or } x\) | \(=\) | \(180^\circ + \alpha \approx \mathbf{250.5^\circ}\) |
Replacing \(x\) by \(x + 30^\circ\) moves the graph \(30^\circ\) to the left. Every key point of \(y = \cos x\) moves back \(30^\circ\).
| \(x\)-intercepts | \(:\) | \(90^\circ - 30^\circ = \mathbf{60^\circ}\), \(270^\circ - 30^\circ = \mathbf{240^\circ}\) |
| minimum \((-1)\) | \(:\) | \(180^\circ - 30^\circ = \mathbf{150^\circ}\) |
| maximum \((1)\) | \(:\) | \(360^\circ - 30^\circ = \mathbf{330^\circ}\) |
The maximum at \(0^\circ\) moves to \(-30^\circ\), outside the domain, so the next one, at \(330^\circ\), is used. Check: \(\cos(330^\circ + 30^\circ) = \cos 360^\circ = 1\).
First find where the curve meets the line \(y = 1\): \(\tan x = 1\) gives \(x = 45^\circ\) or \(180^\circ + 45^\circ = 225^\circ\).
The curve is above the line from each meeting point up to the next asymptote, where \(\tan x\) is undefined:
| \(\) | \(\) | \(\mathbf{45^\circ} < x < \mathbf{90^\circ}\) or \(\mathbf{225^\circ} < x < \mathbf{270^\circ}\) |
The ends are not included: \(\tan x = 1\) there, and \(90^\circ\) and \(270^\circ\) have no value at all.
(i) Find the lowest and highest temperatures and when they occur.
(ii) Between which times is the temperature at least \(21^\circ\)C?
(i) \(-1 \le \cos(15t)^\circ \le 1\), so \(T\) runs from \(18 - 6 = 12\) to \(18 + 6 = 24\).
| lowest \(\mathbf{12^\circ}\)C | \(:\) | \(\cos(15t)^\circ = 1\), \(15t = 0,\ t = 0\) (midnight) |
| highest \(\mathbf{24^\circ}\)C | \(:\) | \(\cos(15t)^\circ = -1\), \(15t = 180,\ t = 12\) (noon) |
(ii) Solve \(T = 21\) first.
| \(18 - 6\cos(15t)^\circ\) | \(=\) | \(21\) |
| \(\cos(15t)^\circ\) | \(=\) | \(-\dfrac{1}{2}\) |
| \(15t\) | \(=\) | \(120 \text{ or } 240\) |
| \(t\) | \(=\) | \(8 \text{ or } 16\) |
The graph is above \(T = 21\) between these times: from \(\mathbf{8}\) am to \(\mathbf{4}\) pm.
Common pitfalls
Frequently asked questions
What does the graph of y = sin x look like?
It is a smooth wave between -1 and 1 that repeats every 360 degrees. It starts at 0, peaks at 1 at 90 degrees, crosses the axis at 180 degrees, dips to -1 at 270 degrees and returns to 0 at 360 degrees.
What is the difference between the sine and cosine graphs?
They have the same wave shape and range, but the cosine graph is shifted 90 degrees to the left. Cosine starts at its maximum of 1 when x is 0, while sine starts at 0.
Why does tan x have asymptotes?
tan x equals sin x divided by cos x. Where cos x is 0, at 90 and 270 degrees, the division is undefined and the graph shoots off to very large positive or negative values, so it has vertical asymptotes there.
What does a do in y = a sin x?
It stretches the graph vertically by a factor of a, so the maximum becomes a and the minimum becomes -a. The x-intercepts do not move.
What does b do in y = sin bx?
It squashes the graph horizontally, so there are b full waves in 360 degrees. Each wave is 360 divided by b degrees long.
What are the amplitude and period of a trig graph?
The period is the length of one full wave: \(360^\circ\) for \(y = \sin x\) and \(y = \cos x\), and \(\dfrac{360^\circ}{b}\) for \(y = \sin bx\). The amplitude is the height from the centre line to a maximum, which is \(a\) for \(y = a\sin x\).