Trig Graphs (AQA GCSE Maths)

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°”

• 
• Starts at (0,0)
• Every 90° it cycles through 1, 0, -1, 0, ...

• 
• Starts at (0, 1)
• Every 90° it cycles through 0, -1, 0, 1, ...

• 
• 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

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

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

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