created by Mr. Lafferty Graphs of the form y = a sin x o Trigonometry Graphs National 5 Graphs of the form y = a sin bx o Solving Trig Equations Special trig relationships Exact values for Sin Cos and Tan Angles greater than 90 o Graphs of the form y = a sin bx o + c

9-Aug-15Created by Mr Lafferty Maths Dept Starter Questions National 5

9-Aug-15Created by Mr Lafferty Maths Dept Exact Values Learning Intention Success Criteria 1.Recognise basic triangles and exact values for sin, cos and tan 30 o, 45 o, 60 o. 1.To build on basic trigonometry values. 2.Calculate exact values for problems. National 5

º º 33 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin, Cos and Tan are useful left as fractions, We call these exact values National 5

x0º30º45º60º90º Sin xº Cos xº Tan xº  ½ ½ 33  3 2  Exact Values National 5

Exact Values º 2 2 Consider the square with sides 1 unit 1 1 We are now in a position to calculate exact values for sin, cos and tan of 45 o National 5

x0º30º45º60º90º Sin xº Cos xº Tan xº  ½ ½ 33  3 2  Exact Values National 5 1  2 1

9-Aug-15Created by Mr Lafferty Maths Dept Now try Ex 2.1 Ch11 (page 220) Exact Values National 5

9-Aug-15Created by Mr Lafferty Maths Dept Starter Questions National 5

9-Aug-15Created by Mr. Lafferty Maths Dept. Learning Intention Success Criteria Angles Greater than 90 o 1.Introduce definition of sine, cosine and tangent over 360 o using triangles with the unity circle. 1.Find values of sine, cosine and tangent over the range 0 o to 360 o. 2.Recognise the symmetry and equal values for sine, cosine and tangent. National 5

9-Aug-15www.mathsrevision.com 11 x y r Angles Greater than 90 o We will now use a new definition to cater for ALL angles. O x-axis r y-axis y x A o New Definitions P(x,y) National 5 Demo SinDemo CosDemo Tan

9-Aug-15Created by Mr Lafferty Maths Dept Trigonometry Angles over 90 0 (1.2, 1.6) 53 o The radius line is 2cm. The point (1.2, 1.6). Find sin cos and tan for the angle. Check answer with calculator Example National 5

9-Aug-15Created by Mr Lafferty Maths Dept Trigonometry Angles over 90 0 The radius line is 2cm. The point (-1.8, 0.8). Find sin cos and tan for the angle. Check answer with calculator Example 1 National 5 (-1.8, 0.8) 127 o

9-Aug-15Created by Mr Lafferty Maths Dept Trigonometry All Quadrants Calculate the ration for sin cos and tan for the angle values below. Summary of results Example National 5 All +veSin +ve Tan +ve Cos +ve 180 o - x o 180 o + x o 360 o - x o 30 o 60 o 150 o 120 o 45 o 135 o 210 o 240 o 330 o 300 o 225 o 315 o Sin xCos xTan x 90 o 180 o 270 o 0 o x o

1)Sin 135 o 2)Cos 150 o 3)Tan 135 o 4)Sin 225 o 5)Cos 270 o What Goes In The Box ? Write down the equivalent values of the following in term of the first quadrant (between 0 o and 90 o ): sin 45 o 1)Sin 300 o 2)Cos 360 o 3)Tan 330 o 4)Sin 380 o 5)Cos 460 o -cos 45 o -tan 45 o -sin 45 o -cos 90 o - sin 60 o cos 0 o - tan 30 o sin 20 o - cos 80 o National 5

9-Aug-15Created by Mr Lafferty Maths Dept Now try MIA Ch11 Ex3.1 Ch11 (page 222) Trigonometry Angles over 90 0 National 5

created by Mr. Lafferty Starter National 5

created by Mr. Lafferty Learning Intention Success Criteria 1.Identify the key points for various graphs. 1.To investigate graphs of the form y = a sin x o y = a cos x o y = tan x o Sine Graph National 5

created by Mr. Lafferty Sine Graph Key Features Domain is 0 to 360 o (repeats itself every 360 o ) Maximum value of 1 Minimum value of -1 Key Features Zeros at 0, 180 o and 360 o Max value at x = 90 o Minimum value at x = 270 o National 5

created by Mr. Lafferty Sine Graph o 180 o 270 o 360 o y = sinx o y = 2sinx o y = 3sinx o y = 0.5sinx o y = -sinx o National 5 What effect does the number at the front have on the graphs ? What effect does the negative sign have on the graphs ? Demo

created by Mr. Lafferty Sine Graph y = a sin (x) 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. National 5

created by Mr. Lafferty Sine Graph o 180 o 270 o 360 o y = 5sinx o y = 4sinx o y = sinx o y = -6sinx o National 5

created by Mr. Lafferty Cosine Graphs Key Features Key Features Domain is 0 to 360 o (repeats itself every 360 o ) Maximum value of 1 Minimum value of -1 Zeros at 90 o and 270 o Max value at x = 0 o and 360 o Minimum value at x = 180 o National 5

created by Mr. Lafferty Cosine o 180 o 270 o 360 o y = cosx o y = 2cosx o y = 3cosx o y = 0.5cosx o y = -cosx o National 5 What effect does the number at the front have on the graphs ? Demo

created by Mr. Lafferty Cosine Graph o 180 o 270 o 360 o y = 2cosx o y = 4cosx o y = 6cosx o y = 0.5cosx o y = -cosx o National 5

created by Mr. Lafferty Tangent Graphs Key Features Key Features Domain is 0 to 180 o (repeats itself every 180 o ) Zeros at 0 and 180 o National 5

created by Mr. Lafferty Tangent Graphs National 5 Demo

created by Mr. Lafferty Tangent Graph y = a tan (x) 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. National 5

created by Mr. Lafferty When a pattern repeats itself over and over, it is said to be periodic. Sine function has a period of 360 o Period of a Function Let’s investigate the function y = sin bx National 5

created by Mr. Lafferty Sine Graph o 180 o 270 o 360 o y = sinx o y = sin2x o y = sin4x o y = sin0.5x o National 5 What effect does the number in front of x have on the graphs ? Demo

created by Mr. Lafferty Trigonometry Graphs y = a sin (bx) 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. How many times it repeats itself in 360 o National 5

created by Mr. Lafferty Cosine o 180 o 270 o 360 o y = cosx o y = cos2x o y = cos3x o National 5 Demo

created by Mr. Lafferty Trigonometry Graphs y = a cos (bx) 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. How many times it repeats itself in 360 o National 5

created by Mr. Lafferty Trigonometry Graphs y = a tan (bx) 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. How many times it repeats itself in 180 o National 5 Demo

y = 0.5sin2x o y = 2sin4x o y = -3sin0.5x o created by Mr. Lafferty Trig Graph o 180 o 270 o 360 o Write down the equations for the graphs shown ? Combinations National 5

created by Mr. Lafferty Cosine o 180 o 270 o 360 o Combinations y = 1.5cos2x o y = -2cos2x o y = 0.5cos4x o National 5 Write down equations for the graphs shown?

created by Mr. Lafferty Now Try MIA Ch11 Ex 5.1 Page Combination Graphs National 5

created by Mr. Lafferty Trigonometry Graphs y = a sin (bx) + c 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. How many times it repeats itself in 360 o National 5 a - Amplitude C moves the graph up or down in the y-axis direction Demo

created by Mr. Lafferty Sine Graph o 180 o 270 o 360 o Simply move graph up by 1 45 o National 5 Given the basic y = sin x graph what does the graph of y = sin x + 1 look like?

created by Mr. Lafferty Cosine Graph o 180 o 270 o 360 o Simply move down by o National 5 Given the y = cos x graph. What does the graph of y = cos x – 0.5 look like?

created by Mr. Lafferty Trig Graph o 180 o 270 o 360 o Write down equations for graphs shown ? Combinations National 5 y = 0.5sin2x o y = 2sin4x o - 1

created by Mr. Lafferty Cosine o 180 o 270 o 360 o Combinations y = cos2x o + 1 y = -2cos2x o - 1 National 5 Write down equations for the graphs shown?

created by Mr. Lafferty Now try MIA Ch11 Ex 5.2 Page Combination Graphs National 5

created by Mr. Lafferty Starter National 5

created by Mr. Lafferty Learning Intention Success Criteria 1.Use the rule for solving any ‘ normal ‘ equation 2.Realise that there are many solutions to trig equations depending on domain. 1.To explain how to solve trig equations of the form a sin x o + 1 = 0 Solving Trig Equations National 5

created by Mr. Lafferty Solving Trig Equations All +veSin +ve Tan +ve Cos +ve 180 o - x o 180 o + x o 360 o - x o 1234 National 5

created by Mr. Lafferty Solving Trig Equations a sin x o + b = 0 Example 1 : Solving the equation sin x o = 0.5 in the range 0 o to 360 o Graphically what are we trying to solve x o = sin -1 (0.5) x o = 30 o There is another solution x o = 150 o (180 o – 30 o = 150 o ) sin x o = (0.5) 1234 National 5

created by Mr. Lafferty Solving Trig Equations a sin x o + b = 0 Example 1 : Solving the equation 3sin x o + 1= 0 in the range 0 o to 360 o Graphically what are we trying to solve sin x o = -1/3 Calculate first Quad value x o = 19.5 o x = 180 o o = o ( 360 o o = o ) There is another solution 1234 National 5

created by Mr. Lafferty Solving Trig Equations a cos x o + b = 0 Example 1 : Solving the equation cos x o = in the range 0 o to 360 o Graphically what are we trying to solve cos x o = x o = 51.3 o (360 o o = o ) x o = cos There is another solution 1234 National 5

created by Mr. Lafferty Solving Trig Equations a tan x o + b = 0 Example 1 : Solving the equation tan x o = 2 in the range 0 o to 360 o Graphically what are we trying to solve tan x o = 2 x o = 63.4 o x = 180 o o = o x o = tan -1 (2) There is another solution 1234 National 5

created by Mr. Lafferty Now try MIA Ch11 Ex6.1, 6.2 and 7.1 (page 236) Solving Trig Equations National 5

created by Mr. Lafferty Starter National 5

created by Mr. Lafferty Learning Intention Success Criteria 1.Know and learn the two special trig relationships. 2.Apply them to solve problems. 1.To explain some special trig relationships sin 2 x o + cos 2 x o = ? and tan x o and sin x cos x Solving Trig Equations National 5

created by Mr. Lafferty Solving Trig Equations Lets investigate sin 2 x o + cos 2 x o = ? Calculate value for x = 10, 20, 50, 250 sin 2 x o + cos 2 x o = 1 Learn ! National 5

created by Mr. Lafferty Solving Trig Equations Lets investigate tan x o Calculate value for x = 10, 20, 50, 250 Learn ! sin x o cos x o and tan x o sin x o cos x o = National 5

created by Mr. Lafferty Now try MIA Ex8.1 Ch11 (page 238) Solving Trig Equations National 5