Trig Graphs

Drawing Trig Graphs

Why do we need to know what trig graphs look like?

Trigonometric graphs (trig graphs) are used in various applications of mathematics
- for example, the oscillating nature can be used to model how a pendulum swings or tide heights

How do we draw trig graphs?

As with other graphs, being familiar with the general style of trigonometric graphs will help you sketch them quickly and you can then use them to find values or angles alongside, or instead of, your calculator
All trigonometric graphs follow a pattern – a “starting point” and then “something happens every 90°”

$y = \sin x$
Starts at (0,0)
Every 90° it cycles through 1, 0, -1, 0, ...

y-equals-sinx

$y = \cos x$
Starts at (0, 1)
Every 90° it cycles through 0, -1, 0, 1, ...

y-equals-cosx

$y = \tan x$
Starts at (0, 0)
Every 90° it is either 0 or an asymptote
- An asymptote is a line that a graph gets ever closer to, without ever crossing or touching it

y-equals-tanx

Worked example

On the axes provided, sketch the graph of $y = \sin x °$ for $0 \leq x \leq 360$ .

Mark key values on the axes provided; 1, 0 and −1 on the y-axis and 0, 90, 180, 270 and 360 on the x-axis. Try to space them evenly apart

For $y = \sin x °$ , the key knowledge is that it starts at (0, 0) then every 90° it cycles though 1, 0 , −1, 0, ... so mark these points on the axes

Finally, join the points with a smooth line. It is best practice to label the curve with its equation

Solving Trig Equations

You can use the symmetry of trig graphs to find multiple solutions to a trig equation

How are trigonometric equations of the form sin x = k solved?

The solutions to the equation sin x = 0.5 in the range 0° < x < 360° are x = 30° and x = 150°

If you like, check on a calculator that both sin(30) and sin(150) give 0.5

The first solution comes from your calculator (by taking inverse sin of both sides)

x = sin^-1(0.5) = 30°

The second solution comes from the symmetry of the graph y = sin x between 0° and 360°

Sketch the graph
Draw a vertical line from x = 30° to the curve, then horizontally across to another point on the curve, then vertically back to the x-axis again
By the symmetry of the curve, the new value of x is 180° - 30° = 150°

In general, if x° is an acute angle that solves sin x = k, then 180° - x° is the obtuse angle that solves the same equation
If the calculator gives x as a negative value, continue drawing the curve to the left of the x-axis to help

cie-igcse-3-12-2-solving-trig-equations-1

How are trigonometric equations of the form cos x = k solved?

The solutions to the equation cos x = 0.5 in the range 0° < x < 360° are x = 60° and x = 300°

If you like, check on a calculator that both cos(60) and cos(300) give 0.5

The first solution comes from your calculator (by taking inverse cos of both sides)

x = cos^-1(0.5) = 60°

The second solution comes from the symmetry of the graph y = cos x between 0° and 360°

Sketch the graph
Draw a vertical line from x = 60° to the curve, then horizontally across to another point on the curve, then vertically back to the x-axis again
By the symmetry of the curve, the new value of x is 360° - 60° = 300°

In general, if x° is an angle that solves cos x = k, then 360° - x° is another angle that solves the same equation
If the calculator gives x as a negative value, continue drawing the curve to the left of the x-axis to help

cie-igcse-3-12-2-solving-trig-equations-2

How are trigonometric equations of the form tan x = k solved?

The solutions to the equation tan x = 1 in the range 0° < x < 360° are x = 45° and x = 225°

Check on a calculator that both tan(45) and tan(225) give 1

The first solution comes from your calculator (by taking inverse tan of both sides)

x = tan^-1(1) = 45°

The second solution comes from the symmetry of the graph y = tan x between 0° and 360°

Sketch the graph
Draw a vertical line from x = 45° to the curve, then horizontally across to another point on the curve (a different “branch” of tan x), then vertically back to the x-axis again
The new value of x is 45° + 180° = 225° as the next “branch” of tan x is shifted 180° to the right

In general, if x° is an angle that solves tan x = k, then x° + 180° is another angle that solve the same equation
If the calculator gives x as a negative value, continue drawing the curve to the left of the x-axis to help

Exam Tip

Use a calculator to check your solutions by substituting them into the original equation
For example, 60° and 330° are incorrect solutions of cos x = 0.5, as cos(330) on a calculator is not equal to 0.5

Worked example

Solve sin x = 0.25 in the range 0° < x < 360°, giving your answers correct to 1 decimal place

Use a calculator to find the first solution (by taking inverse sin of both sides)

x = sin^-1(0.25) = 14.4775… = 14.48° to 2 dp

Sketch the graph of y = sin x and mark on (roughly) where x = 14.48 and y = 0.25 would be
Draw a vertical line up to the curve, then horizontally across to the next point on the curve, then vertically back down to the x-axis