Nat 5 Trigonometric Functions and Graphs of the

Nat 5 Trigonometric Functions and Graphs of the form y = a sin xo Graphs of the form y = a sin bxo Phase angle Solving Trig Equations Special trig relationships

Trigonometric Functions and Graphs Learning Intention 1. To identify key features of graphs of trigonometric functions including: y = a sin xo y = a cos xo y = a tan xo

Key Features Sine Function Graph Zeros at 0, 180 and 360 o o Maximum value at x = 90 o Minimum value at x = 270 o Key Features Domain is 0 to 360 o (repeats itself every 360 o) Maximum value of 1 Minimum value of -1

Sine Function Graph y = sinxo y = 2 sinxo y = 3 sinxo 3 y = 0. 5 sinxo y = -sinxo 2 1 0 -1 -2 -3 90 o 180 o 270 o 360 o

Sine Function Graph y = 5 sinxo y = 4 sinxo y = sinxo 6 y = -6 sinxo 4 2 0 -2 -4 -6 90 o 180 o 270 o 360 o

Key Features Cosine Function Graph Zeros at 90 and 270 o o Maximum value at x = 0 o and 360 o Minimum value at x = 180 o Key Features Domain is 0 to 360 o (repeats itself every 360 o) Maximum value of 1 Minimum value of -1

y = cosxo Cosine Function Graphy = 2 cosx o y = 3 cosxo 3 y = 0. 5 cosxo 2 y = -cosxo 1 0 -1 -2 -3 90 o 180 o 270 o 360 o

y = cosxo Cosine Function Graph y = 4 cosx o y = 6 cosxo 6 y = 0. 5 cosxo y = -1. 5 cosxo 4 2 0 -2 -4 -6 90 o 180 o 270 o 360 o

Tangent Function Graph Zeros at 0 and 180 Key Features o Key Features Undefined at 90 o and 270 o Key Features Domain is 0 to 180 o (repeats itself every 180 o)

Tangent Function Graph created by Mr. Lafferty

Cosine Function Graph y = a sin (x) y = a cos (x) 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 < 0 graph reflects in the x – axis.

Trigonometric Functions and Graphs Learning Intention 1. To identify key features of graphs of trigonometric functions including: y = sin bxo y = cos bxo y = tan bxo

Period of a Function When a pattern repeats itself over and over, it is said to be periodic. Sine function has a period of 360 o Cosine function has a period of 360 o Consider and y = sin bx y = cos bx

Sine Function Graphy = sinx o y = sin 2 xo y = sin 4 xo 3 y = sin 0. 5 xo 2 1 0 -1 -2 -3 90 o 180 o 270 o 360 o

Cosine Function Graph y = cosxo y = cos 2 xo 3 y = cos 3 xo 2 1 0 -1 -2 -3 90 o 180 o 270 o 360 o

Period of a Function When a pattern repeats itself over and over, it is said to be periodic. Tangent function has a period of 180 o Consider y = tan bx

Tangent Function Graph y = tanxo

Tangent Function Graph y = tan 2 xo

Tangent Function Graph y = tan 3 xo

Sine, Cosine & Tangent Functions y = a sin (bx) y = a cos (bx) y = a tan (bx) b is how many times graph repeats itself in 360 o b is how many times it repeats itself in 180 o

Trigonometric Functions and Graphs Learning Intentions 1. To identify key features of graphs of trigonometric functions including: y = asin bxo y = acos bxo y = atan bxo 2. To sketch graphs of trigonometric functions of this form.

Trigonometric Graphs y = 0. 5 sin 2 x Write down an equation for the graph shown. y = 2 sin 4 xo 3 y = 3 sin 0. 5 xo 2 1 0 -1 -2 -3 o 90 o 180 o 270 o 360 o

Write down an equation for the graph shown. y = 1. 5 cos 2 xo Trigonometric Graphs y = -2 cos 2 x y = 0. 5 cos 4 xo 3 2 1 0 -1 -2 -3 o 90 o 180 o 270 o 360 o

Trigonometric Functions and Graphs Learning Intentions 1. To identify the phase angle in graphs of trigonometric functions of the form: y = a sin (x-b)o 2. To sketch graphs of trigonometric functions of the form: y = a sin (x-b)o

The Sine Function Graph y = sin(x - 45)o 1 0 -1 45 o 90 o 180 o 270 o 360 o

The Sine Function Graph y = sin(x + 60)o 1 60 o -60 o 0 -1 90 o 180 o 270 o 360 o

By how much do we have to move the ‘new’ cosine curve so it fits on the Int 2 original cosxo curve? The Cosine Function Graph y = cos(x - 70)o 1 0 -1 70 o o 90 o 160 180 o 270 o 360 o

By how much do we have to move the ‘new’ cosine curve so it fits on the Int 2 original cosxo curve? The Cosine Function Graph y = cos(x + 56)o 1 0 -1 56 o 34 o 90 o 180 o 270 o 360 o

Phase Angle y = sin (x - b) y = cos (x - b) b moves graph along x – axis.

Naming a Function y = a cos (x – b) a =3 b = -30 y = 3 cos (x - 30)

By how much do we have o curve o and o to move the cosx Similarly, sinx cosx o o cos(x+90) = sinx o o arefits 90 exactly outo of phase. so sin(x-90) it onto the = cosx sinxo curve? Phase Angle & Graphs 1 0 -1 180 o 360 o 540 o 720 o