The graphs of sin  and cos 

Trigonometric Ratios TOA CAH SOH S O HC A H T O A

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

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

Graphs ofand y x y y e.g.

Graphs ofand y x y y e.g.

Graphs ofand y x y y e.g.

Graphs ofand y y e.g. x y

Graphs ofand y x y y e.g.

Graphs ofand y x y Also, when x x x x x x

Graphs ofand x x x x x x x y y

Graphs ofand As increases from to, y increases from 0 to 1. y

Graphs ofand x x e.g. For angles between 90 and 180 the height of the triangle decreases

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

Graphs ofand The graph of between and is

Graphs ofand For angles between and the height of the triangle is negative and so the graph appears under the horizontal axis

Graphs ofand We can extend the graph as shown

Graphs ofand We can extend the graph as shown

Graphs ofand 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 ! We can now draw the graph of for any interval. e.g.

Graphs ofandSUMMARY 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.

Graphs ofandExercises 1. Sketch the graph of for the interval (a) Write down an angle between and ( not equal to the given angle! ) where (b) (a) Ans: xx

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

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

Graphs ofand P (x, y) x y ON x Graph of cos 

Graphs ofand x y O As increases, x decreases from 1 to 0 x = 1 x = 0 e.g. When we sketch the graph we use y instead of x.

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

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

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