Grade 10 Mathematics Trigonometric functions

Table of Contents Draw accurate graphs of the functions and Draw sketch graphs of and using their properties Draw accurate graphs of the function Draw a sketch graph of using its properties. Investigate the influence of a in the graphs of , and Investigate the influence of q on the graphs of , and 7. Use trigonometric graphs to describe real – life situations.

Draw accurate graphs of the functions and Trigonometric functions are used to describe periodic phenomenon. It is always better to set up a table if you are working with an unknown function. 1 - -1

Draw accurate graphs of the functions and Note: The sine curve starts at the origin and resembles a wave-like curve. It takes to complete one cycle and thus the period of the function is The amplitude of the graph is 1, that is the biggest distance from the position of rest. The Domain is: x Є R and the range is The sin function changes, repeats values over a certain interval and is called a periodic function.

The Graph of the Sine Function Basic sin curve y x

The Graph of the Sine Function y x Basic cos curve

Draw sketch graphs of and using their properties Properties of the sin – function: Period = The sin – function is a periodic function because there is a repetition of values over a certain interval (check table) Amplitude = 1 Range: The graph starts at ( ;0) The sin – function is positive in the first and second quadrant. Use the five points: ,

Draw sketch graphs of and using their properties Properties of the cos – function: Period = The cos function is a periodic function because there is a repetition of values over a certain interval (check table) Amplitude = 1 Range: The graph starts at The values of cos changes as changes from to The cos – function is positive in the first and fourth quadrant. Use the five points: ,

Sin – function: Using the five points plus the sign in the four quads y x

Cos – function: Using the five points plus the sign in the four quads y x

Draw accurate graphs of the function If: x- intercepts are at: Asymptotes at: Amplitude: the tan curve has no amplitude, because it has no maximum or minimum value Period of the tan – function = 180º Range: y Є R Domain: x Є [ 0º ; 360º ] ; x ≠ ± 90º ; x ≠ 270º

Test Your Knowledge On the same set of axis sketch the graphs of: clearly showing the translations that take place.

Solutions y x

The tan - function x

Draw a sketch graph of using its properties. Θ Є [ -90º ; 360º ] y x -

Investigate the influence of a in the graphs of , and In both the sin and the cos function we see a change in amplitude: amplitude = a units period = remains range = y x

The transformations in the graph of the cos – function, due to the influence of a y x

x

y x The following graphs clearly show how the graphs of the sin - and cos functions are translated when q is added to the parent graphs

Use trigonometric graphs to describe real life situations The heights of tides with respect to time are illustrated in the following sketch. The solid line represents the curve of heights of tides against time. What is the value of a and q if the equation of the curve is given by y x The equation is:

Test your knowledge Question 1 Write down the amplitude and period of the graph y= - 2 cos2x Answer A Amplitude = -2 Period = 360° B Amplitude = 2 Period = 180° C Amplitude = -2 Period = 360° D None of the above

Test your Knowledge Question 2 Write down the amplitude and period of the graph y = sin x – 2 Answer A Amplitude = 1 Period = 360° B Amplitude = 2 Period =360° C Amplitude = - 2 Period = 180° D None of the above

Test your knowledge Question 3 Write down the range and period of the graph y = 2 tanx Answer A Range : y ε R Period = 360° B Range : y ε [ - 2 ; 2 ] Period = 360° C Range : y ε R Period = 180° D None of the above

Test your knowledge Question 4 Write down the range of the graph y = 2sinx – 1 Answer A Range : Range : y ε R B Range : y ε [ - 2 ; 2 ] C Range y ε [ - 1; 3 ] D Range : y ε [ - 3; 1 ]

Test your knowledge Question 5 Write down the amplitude of the graph y = 3tan2x Answer A Amplitude = 3 B Amplitude = 2 C No Amplitude D None of the above

Problems 1. On a certain day the depth, D metres, of water at a fishing port, t hours after midnight, is given by a. Find the depth of the water at 1.30 pm. b. The depth of the water in the harbour is recorded each hour. What is the maximum difference in the depths of the water over the 24 hour period?

Solution

Problem Solve

Solution

Trig Graph collection The following slides give more exemplars of trig graphs. Please study them

90o The Trigonometric Ratios for any angle 180o 0o 270o 1 0o 90o 180o 90 180 360 270 -90 -180 -270 -360 90 180 270 -90 -180 -270 -360 360 The Trigonometric Ratios for any angle 180o 0o 450o 0o 90o 180o 270o 360o -90o -180o -270o -360o -450o 270o 1 0o 90o 180o 270o 360o  -1

-360 90 180 x y = f(x) 360 -90 -180 -270 1 -1 2 -2 sinx 2sinx 3 -3 3sinx y = ½sinx 270

Period 360o 3 Amplitude  3 3Sinx 2Sinx Period 360o 2 Amplitude  2 Sinx Period 360o 1 Amplitude  1 x -270 -180 -90 90 180 270 360 -360 -1 -2 -3

90 180 x y = f(x) 270 360 -90 -180 -270 -360 1 -1 2 -2 3 -3 cosx ½cosx 2cosx 3cosx

y = sinx y = 2sinx x x y = 3sinx y = ½ sinx x x 3 3 2 2 1 1 - 1 - 1 1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o 1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o x x y = 3sinx y = ½ sinx 1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o 1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o x x

y = cosx y = 2cosx x x y = 3cosx y = ½ cosx x x 3 3 2 2 1 1 - 1 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o 1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o x x y = 3cosx y = ½ cosx 1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o 3 2 1 x x -360o -270o -180o -90o 90o 180o 270o 360o - 1 - 2 - 3

-360 y = f(x) 90 180 -90 -180 -270 1 -1 2 -2 f(x) = sinx f(x) = sin2x f(x) = sin3x f(x) = sin ½ x x 270 360

x 270 360 90 -360 180 y = f(x) -90 -180 -270 1 -1 2 -2 f(x) = cosx f(x) = cos2x f(x) = cos3x f(x) = cos ½ x

y = sinx y = sin2x x x y = sin3x y = sin ½ x x x 3 3 2 2 1 1 - 1 - 1 1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o 1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o x x y = sin3x y = sin ½ x 1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o 1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o x x

y =cosx y = cos2x x x y = cos3x y = cos ½ x x x 3 3 2 2 1 1 - 1 - 1 1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o 1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o x x y = cos3x y = cos ½ x 1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o 3 2 1 x x -360o -270o -180o -90o 90o 180o 270o 360o - 1 - 2 - 3

y = tan  -450o -360o x -270o -180o -90o 0o 90o 180o 270o 360o 450o

y = sinx y = cosx x x y = tan  x 1 2 3 - 1 - 2 - 3 1 2 3 - 1 - 2 - 3 1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o 1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o x x y = tan  x -450o -360o -270o -180o -90o 0o 90o 180o 270o 360o 450o

x x y = 2cosx y = sinx y = cosx y = 2sinx y = 3sinx y = ½ sinx 1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o 1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o x x y = 3sinx y = 3cosx y = ½ sinx y = ½ cosx 1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o 1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o x x

y = sinx y = - sinx y = 2sinx y = -2sinx x x y = 3sinx y = - 3sinx 1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o 1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o x x y = 3sinx y = - 3sinx y = ½ sinx y = - ½ sinx 1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o 1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o x x

y = cosx y = -cosx y = 2cosx y = -2cosx x x y = 3cosx y = -3cosx 1 1 x x -360o -270o -180o -90o 90o 180o 270o 360o -360o -270o -180o -90o 90o 180o 270o 360o - 1 - 1 - 2 - 2 - 3 - 3 y = 3cosx y = -3cosx y = ½ cosx y = -½ cosx 3 3 2 2 1 1 -360o x x -270o -180o -90o 90o 180o 270o 360o -360o -270o -180o -90o 90o 180o 270o 360o - 1 - 1 - 2 - 2 - 3 - 3

270 -360 90 180 x y = f(x) 360 -90 -180 -270 1 -1 2 -2 3 -3

1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o x 1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o 1 2 3 - 1 - 2 - 3 90o 180o 270o 360o -90o -180o -270o -360o