37: The graphs of sinq and cosq “Teach A Level Maths” Vol. 1: AS Core Modules 37: The graphs of sinq and cosq © Christine Crisp

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Trig Functions and Graphs We are going to sketch the graph of where is an angle between and .

Let P be a point with coordinates (x, y) on the circle with centre at the origin and radius 1. P (x, y) x y O Let angle PON be Then, y N

e.g. x y y 1 20 40 60 80 y

e.g. x y y 1 20 40 60 80 y

e.g. x y y 1 20 40 60 80 y

e.g. x y y 1 20 40 60 80 y

e.g. x y y 1 20 40 60 80 y

Also, when x y y 1 x x x x x x 20 40 60 80

x y y 1 20 40 60 80 x x x x x x

As increases from to , y increases from 0 to 1. 20 40 60 80 10 30 50 70 As increases from to , y increases from 0 to 1.

We have, so far, only defined for angles between and . We will now extend the definitions for an angle which can be any size and may be positive or negative.

Suppose , then x y

e.g. suppose , then x y

If , . . . x y

If , . . . x y . . . then y is still y But So,

The quarter of the circle where the angles are between and is called the 1st quadrant. For each angle in the 1st quadrant there is an angle of in the 2nd quadrant where

Notice the symmetry about the y-axis . e.g. x y Notice the symmetry about the y-axis .

The symmetry about enables us to draw the graph for between and . x e.g.

The symmetry about enables us to draw the graph for between and . x e.g.

The graph of between and is

For angles between and ( the 3rd and 4th quadrants ), the y coordinates are negative. x y e.g. So,

The sine of each angle between and has the same magnitude as an angle between and BUT is negative. The graph for is:

For angles larger than we continue round the circle, so the values of repeat every . e.g. y x

We can extend the graph as shown

We can extend the graph as shown

Negative angles are defined as those formed by turning in a clockwise direction from . x So,

We can now draw the graph of for any interval. e.g. If you have a graphical calculator, this graph will be one of the standard graphs BUT make sure you can also sketch it without your calculator !

SUMMARY The trig function is defined for any angle. The graph of repeats every . The minimum value of is and the maximum is . The graph for is . . . . . . and must be memorised.

1. Sketch the graph of for the interval Exercises 1. Sketch the graph of for the interval Write down an angle between and ( not equal to the given angle! ) where (a) (b) (c) (d) Ans: (a) x x

1. Sketch the graph of for the interval Exercises 1. Sketch the graph of for the interval Write down an angle between and ( not equal to the given angle! ) where (a) (b) (c) (d) Ans: (a) (b) x x

1. Sketch the graph of for the interval Exercises 1. Sketch the graph of for the interval Write down an angle between and ( not equal to the given angle! ) where (a) (b) (c) (d) Ans: (a) x (b) x x (c)

1. Sketch the graph of for the interval Exercises 1. Sketch the graph of for the interval Write down an angle between and ( not equal to the given angle! ) where (a) (b) (c) (d) Ans: (a) x (b) x (c) (d)

Exercises Sketch a circle with radius 1 and centre at the origin O. Mark a point P (x, y) on the circumference in the 1st quadrant. Using the sketch, write down an expression for . 2(a) (b) Mark , the angle between the radius OP and the x-axis. (c) (d) (f) Refer to the circle to determine symmetry and hence complete the graph for . Use the sketch to estimate enough values of to sketch the graph of for . (e)

Exercises P (x, y) x y O N 2. Solution:

As increases, x decreases from 1 to 0 Exercises y 2. Solution: x = 0 e.g. x x = 1 O As increases, x decreases from 1 to 0 When we sketch the graph we use y instead of x.

Exercises 2. Solution: Notice that is symmetric about ( the y-axis ).

The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

The quarter of the circle where the angles are between and is called the 1st quadrant. For each angle in the 1st quadrant there is an angle of in the 2nd quadrant where o ) 180 ( sin q - =

Notice that is symmetric about ( the y-axis ).

The trig functions and are defined for any angle. The graphs repeat every . The minimum value is and the maximum is . The graphs for are