Trigonometry Graphs www.mathsrevision.com S4 Credit Exact values for Sin Cos and Tan Angles greater than 90o Graphs of the form y = a sin xo Graphs of the form y = a sin bxo www.mathsrevision.com Solving Trig Equations Special trig relationships created by Mr. Lafferty

Created by Mr Lafferty Maths Dept Starter Questions S4 Credit www.mathsrevision.com 28-May-18 Created by Mr Lafferty Maths Dept

Created by Mr Lafferty Maths Dept Exact Values S4 Credit Learning Intention Success Criteria To build on basic trigonometry values. Recognise basic triangles and exact values for sin, cos and tan 30o, 45o, 60o . Calculate exact values for problems. www.mathsrevision.com 28-May-18 Created by Mr Lafferty Maths Dept

This triangle will provide exact values for S4 Credit Some special values of Sin, Cos and Tan are useful left as fractions, We call these exact values 60º 2 2 60º 30º 3 www.mathsrevision.com 1 This triangle will provide exact values for sin, cos and tan 30º and 60º

Exact Values ½  ½ 1 1 x 0º 30º 45º 60º 90º www.mathsrevision.com 3 S4 Credit x 0º 30º 45º 60º 90º Sin xº Cos xº Tan xº  3 2 ½ 1 www.mathsrevision.com 3 2 1 ½ 3

Exact Values 45º 2 1 1 45º 1 1 www.mathsrevision.com S4 Credit Consider the square with sides 1 unit 45º 2 1 1 www.mathsrevision.com 45º 1 1 We are now in a position to calculate exact values for sin, cos and tan of 45o

Exact Values ½  ½ 1 1 1 x 0º 30º 45º 60º 90º 1 2 S5 Int2 x 0º 30º 45º 60º 90º Sin xº Cos xº Tan xº  3 2 ½ 1 2 1 www.mathsrevision.com 3 2 1 2 1 ½ 1 3

Created by Mr Lafferty Maths Dept Exact Values S5 Int2 Now try Ex 2.1 Ch11 (page 220) www.mathsrevision.com 28-May-18 Created by Mr Lafferty Maths Dept

Created by Mr Lafferty Maths Dept Starter Questions S4 credit www.mathsrevision.com 28-May-18 Created by Mr Lafferty Maths Dept

Created by Mr. Lafferty Maths Dept. Angles Greater than 90o S4 credit Learning Intention Success Criteria Introduce definition of sine, cosine and tangent over 360o using triangles with the unity circle. Find values of sine, cosine and tangent over the range 0o to 360o. 2. Recognise the symmetry and equal values for sine, cosine and tangent. www.mathsrevision.com 28-May-18 Created by Mr. Lafferty Maths Dept.

Angles Greater than 90o www.mathsrevision.com r r y x 5/28/2018 x y r Angles Greater than 90o S4 credit We will now use a new definition to cater for ALL angles. New Definitions y-axis y x P(x,y) www.mathsrevision.com r We start by find the equation of a circle centre the origin. First draw set axises x,y and then label the origin O. Next we plot a point P say, which as coordinates x,y. Next draw a line from the origin O to the point P and label length of this line r. If we now rotate the point P through 360 degrees keep the Origin fixed we trace out a circle with radius r and centre O. Remembering Pythagoras’s Theorem from Standard grade a square plus b squared equal c squares we can now write down the equal of any circle with centre the origin. Ao O x-axis 28-May-18 www.mathsrevision.com www.mathsrevision.com

Trigonometry www.mathsrevision.com 53o Angles over 900 S4 credit Example The radius line is 2cm. The point (1.2, 1.6). Find sin cos and tan for the angle. (1.2, 1.6) Check answer with calculator www.mathsrevision.com 53o 28-May-18 Created by Mr Lafferty Maths Dept

Trigonometry www.mathsrevision.com 127o Angles over 900 Example 1 S4 credit Check answer with calculator The radius line is 2cm. The point (-1.8, 0.8). Find sin cos and tan for the angle. (-1.8, 0.8) www.mathsrevision.com 127o 28-May-18 Created by Mr Lafferty Maths Dept

Created by Mr Lafferty Maths Dept Summary of results Trigonometry All Quadrants Example S4 credit Calculate the ration for sin cos and tan for the angle values below. 90o 30o 210o 45o 225o Sin +ve All +ve 60o 240o 180o - xo xo 120o 300o www.mathsrevision.com 180o 0o 135o 315o 180o + xo 360o - xo 150o 330o Tan +ve Cos +ve Sin x Cos x Tan x 28-May-18 Created by Mr Lafferty Maths Dept 270o

What Goes In The Box ? www.mathsrevision.com S4 credit Write down the equivalent values of the following in term of the first quadrant (between 0o and 90o): Sin 135o Cos 150o Tan 135o Sin 225o Cos 270o Sin 300o Cos 360o Tan 330o Sin 380o Cos 460o sin 45o - sin 60o -cos 45o cos 0o www.mathsrevision.com -tan 45o - tan 30o -sin 45o sin 20o -cos 90o - cos 80o

Now try MIA Ch11 Ex3.1 Ch11 (page 222) Trigonometry Angles over 900 S4 credit Now try MIA Ch11 Ex3.1 Ch11 (page 222) www.mathsrevision.com 28-May-18 Created by Mr Lafferty Maths Dept

Starter S4 Credit www.mathsrevision.com created by Mr. Lafferty

Sine Graph www.mathsrevision.com Learning Intention Success Criteria S4 Credit Learning Intention Success Criteria To investigate graphs of the form y = a sin xo y = a cos xo y = tan xo Identify the key points for various graphs. www.mathsrevision.com created by Mr. Lafferty

Sine Graph www.mathsrevision.com Key Features Zeros at 0, 180o and 360o Max value at x = 90o S4 Credit Minimum value at x = 270o Key Features www.mathsrevision.com Domain is 0 to 360o (repeats itself every 360o) Maximum value of 1 Minimum value of -1 created by Mr. Lafferty

Sine Graph www.mathsrevision.com 3 2 1 -1 -2 -3 y = sinxo y = 2sinxo What effect does the number at the front have on the graphs ? y = sinxo y = 2sinxo y = 3sinxo y = 0.5sinxo y = -sinxo Sine Graph S4 Credit 3 2 1 www.mathsrevision.com 90o 180o 270o 360o -1 What effect does the negative sign have on the graphs ? -2 -3 created by Mr. Lafferty

Sine Graph y = a sin (x) www.mathsrevision.com S4 Credit y = a sin (x) www.mathsrevision.com For a > 1 stretches graph in the y-axis direction For a < 1 compresses graph in the y - axis direction For a - negative flips graph in the x – axis. created by Mr. Lafferty

Sine Graph www.mathsrevision.com 6 4 2 -2 -4 -6 y = 5sinxo y = 4sinxo S4 Credit 6 4 2 www.mathsrevision.com 90o 180o 270o 360o -2 -4 -6 created by Mr. Lafferty

Cosine Graphs www.mathsrevision.com Key Features Zeros at 0, 90o and 270o Max value at x = 0o and 360o S4 Credit Minimum value at x = 180o Key Features www.mathsrevision.com Domain is 0 to 360o (repeats itself every 360o) Maximum value of 1 Minimum value of -1 created by Mr. Lafferty

What effect does the number at the front have on the graphs ? y = cosxo y = 2cosxo y = 3cosxo y = 0.5cosxo y = -cosxo Cosine S4 Credit 3 2 1 www.mathsrevision.com 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty

Cosine Graph www.mathsrevision.com 6 4 2 -2 -4 -6 y = 2cosxo S4 Credit 6 4 2 www.mathsrevision.com 90o 180o 270o 360o -2 -4 -6 created by Mr. Lafferty

Tangent Graphs www.mathsrevision.com Key Features Zeros at 0 and 180o S4 Credit Key Features www.mathsrevision.com Domain is 0 to 180o (repeats itself every 180o) created by Mr. Lafferty

Tangent Graphs S4 Credit www.mathsrevision.com created by Mr. Lafferty

Tangent Graph y = a tan (x) www.mathsrevision.com S4 Credit y = a tan (x) www.mathsrevision.com For a > 1 stretches graph in the y-axis direction For a < 1 compresses graph in the y - axis direction For a - negative flips graph in the x – axis. created by Mr. Lafferty

Period of a Function y = sin bx www.mathsrevision.com S4 Credit When a pattern repeats itself over and over, it is said to be periodic. Sine function has a period of 360o www.mathsrevision.com Let’s investigate the function y = sin bx created by Mr. Lafferty

What effect does the number in front of x have on the graphs ? y = sinxo y = sin2xo y = sin4xo y = sin0.5xo Sine Graph S4 Credit 3 2 1 www.mathsrevision.com 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty

Trigonometry Graphs y = a sin (bx) www.mathsrevision.com S4 Credit y = a sin (bx) How many times it repeats itself in 360o www.mathsrevision.com For a > 1 stretches graph in the y-axis direction For a < 1 compresses graph in the y - axis direction For a - negative flips graph in the x – axis. created by Mr. Lafferty

Cosine www.mathsrevision.com 3 2 1 -1 -2 -3 y = cosxo y = cos2xo S4 Credit 3 2 1 www.mathsrevision.com 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty

Trigonometry Graphs y = a cos (bx) www.mathsrevision.com S4 Credit y = a cos (bx) How many times it repeats itself in 360o www.mathsrevision.com For a > 1 stretches graph in the y-axis direction For a < 1 compresses graph in the y - axis direction For a - negative flips graph in the x – axis. created by Mr. Lafferty

Trigonometry Graphs y = a tan (bx) www.mathsrevision.com S4 Credit y = a tan (bx) How many times it repeats itself in 180o www.mathsrevision.com For a > 1 stretches graph in the y-axis direction For a < 1 compresses graph in the y - axis direction For a - negative flips graph in the x – axis. created by Mr. Lafferty

Write down the equations for the graphs shown ? y = 0.5sin2xo y = 2sin4xo y = -3sin0.5xo Trig Graph Combinations S4 Credit 3 2 1 www.mathsrevision.com 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty

Write down equations for the graphs shown? y = 1.5cos2xo y = -2cos2xo y = 0.5cos4xo Cosine Combinations S4 Credit 3 2 1 www.mathsrevision.com 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty

Combination Graphs Now Try MIA Ch11 Ex 5.1 Page 227 S4 Credit Now Try MIA Ch11 Ex 5.1 Page 227 www.mathsrevision.com created by Mr. Lafferty

Sine Graph www.mathsrevision.com Simply move graph up by 1 1 -1 45o S4 Credit 1 www.mathsrevision.com 45o 90o 180o 270o 360o Given the basic y = sin x graph what does the graph of y = sin x +1 look like? -1 created by Mr. Lafferty

Cosine Graph www.mathsrevision.com Simply move down by 0.5 1 -1 Given the y = cos x graph. What does the graph of y = cos x – 0.5 look like? Cosine Graph Simply move down by 0.5 S4 Credit 1 www.mathsrevision.com 90o 160o 180o 270o 360o -1 created by Mr. Lafferty

Write down equations for graphs shown ? y = 0.5sin2xo + 1 y = 2sin4xo- 1 Trig Graph Combinations S4 Credit 3 2 1 www.mathsrevision.com 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty

Write down equations for the graphs shown? Cosine y = cos2xo + 1 y = -2cos2xo - 1 Combinations S4 Credit 3 2 1 www.mathsrevision.com 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty

Combination Graphs Now try MIA Ch11 Ex 5.2 Page 227 S4 Credit Now try MIA Ch11 Ex 5.2 Page 227 www.mathsrevision.com created by Mr. Lafferty

Starter S4 Credit www.mathsrevision.com created by Mr. Lafferty

Solving Trig Equations S4 Credit Learning Intention Success Criteria To explain how to solve trig equations of the form a sin xo + 1 = 0 Use the rule for solving any ‘ normal ‘ equation Realise that there are many solutions to trig equations depending on domain. www.mathsrevision.com created by Mr. Lafferty

Solving Trig Equations S4 Credit 1 2 3 4 Sin +ve All +ve 180o - xo 180o + xo 360o - xo www.mathsrevision.com Tan +ve Cos +ve created by Mr. Lafferty

Solving Trig Equations Graphically what are we trying to solve a sin xo + b = 0 S4 Credit Example 1 : Solving the equation sin xo = 0.5 in the range 0o to 360o sin xo = (0.5) 1 2 3 4 xo = sin-1(0.5) www.mathsrevision.com xo = 30o There is another solution xo = 150o (180o – 30o = 150o) created by Mr. Lafferty

Solving Trig Equations Graphically what are we trying to solve a sin xo + b = 0 S4 Credit Example 1 : Solving the equation 3sin xo + 1= 0 in the range 0o to 360o 1 2 3 4 sin xo = -1/3 www.mathsrevision.com Calculate first Quad value xo = 19.5o x = 180o + 19.5o = 199.5o There is another solution ( 360o - 19.5o = 340.5o) created by Mr. Lafferty

Solving Trig Equations Graphically what are we trying to solve a cos xo + b = 0 S4 Credit Example 1 : Solving the equation cos xo = 0.625 in the range 0o to 360o 1 2 3 4 cos xo = 0.625 xo = cos -1 0.625 www.mathsrevision.com xo = 51.3o There is another solution (360o - 53.1o = 308.7o) created by Mr. Lafferty

Solving Trig Equations Graphically what are we trying to solve a tan xo + b = 0 S4 Credit Example 1 : Solving the equation tan xo = 2 in the range 0o to 360o 1 2 3 4 tan xo = 2 xo = tan -1(2) www.mathsrevision.com xo = 63.4o There is another solution x = 180o + 63.4o = 243.4o created by Mr. Lafferty

Solving Trig Equations S4 Credit Now try MIA Ch11 Ex6.1, 6.2 and 7.1 (page 236) www.mathsrevision.com created by Mr. Lafferty

Starter S4 Credit www.mathsrevision.com created by Mr. Lafferty

Solving Trig Equations S4 Credit Learning Intention Success Criteria To explain some special trig relationships sin 2 xo + cos 2 xo = ? and tan xo and sin x cos x Know and learn the two special trig relationships. Apply them to solve problems. www.mathsrevision.com created by Mr. Lafferty

Solving Trig Equations S4 Credit Lets investigate sin 2xo + cos 2 xo = ? Calculate value for x = 10, 20, 50, 250 www.mathsrevision.com sin 2xo + cos 2 xo = 1 Learn ! created by Mr. Lafferty

Solving Trig Equations S4 Credit Lets investigate sin xo cos xo tan xo and Calculate value for x = 10, 20, 50, 250 www.mathsrevision.com sin xo cos xo tan xo = Learn ! created by Mr. Lafferty

Solving Trig Equations S4 Credit Now try MIA Ex8.1 Ch11 (page 238) www.mathsrevision.com created by Mr. Lafferty