Tso math trig graphs

Trigonometry Graphs
S4 Credit

Exact values for Sin Cos and Tan
www.mathsrevision.com

Angles greater than 90o
Graphs of the form y = a sin xo

Graphs of the form y = a sin bxo
Graphs of the form y = a sin bxo + c
Solving Trig Equations

Special trig relationships
created by Mr. Lafferty

Starter Questions
S4 Credit

1. Factorise x2 - 36

2. A car depreciates at 20% each year.
How much is it worth after 1 year if it cost
£15 000 initially.

3. What is sin30o as a fraction.

23 Mar 2013 Created by Mr Lafferty Maths Dept

Exact Values
S4 Credit

Learning Intention Success Criteria

1. To build on basic 1. Recognise basic triangles and
trigonometry values. exact values for sin, cos and
tan 30o, 45o, 60o .

2. Calculate exact values for
problems.


Exact Values
S4 Credit

Some special values of Sin, Cos and Tan are useful
left as fractions, We call these exact values

60º
30º
2 2 2
√3

60º 60º 60º
2 1
This triangle will provide exact values for
sin, cos and tan 30º and 60º

Exact Values
S4 Credit

x 0º 30º 45º 60º 90º

√3
Sin xº 0 ½ 2
1
√3
Cos xº 1 2
½ 0
1
Tan xº 0
3
√3 ∞

Exact Values
S4 Credit

Consider the square with sides 1 unit

45º
1 1
√2
45º
1 1
We are now in a position to calculate
exact values for sin, cos and tan of 45o

Exact Values
S5 Int2

x 0º 30º 45º 60º 90º

1 √3
Sin xº 0 ½ √2 2
1
√3 1
Cos xº 1 2 √2 ½ 0
1
Tan xº 0
3 1 √3 ∞

Exact Values
S5 Int2

Now try Ex 2.1
Ch11 (page 220)


Starter Questions
S4 credit

1. True or false 2 + 3 × 7 = 35

2. A house increases by 3% each year.
How much is it worth after 1 years if it cost
£40 000 initially.

3. What is the exact value of sin 45o.


Angles Greater than 90o
S4 credit


1. Introduce definition of 1. Find values of sine, cosine
sine, cosine and tangent and tangent over the range 0o
over 360o using triangles to 360o.
with the unity circle.
2. Recognise the symmetry and
equal values for sine, cosine
and tangent.

23 Mar 2013 Created by Mr. Lafferty Maths Dept.

r
y Angles Greater than 90o
x
S4 credit

We will now use a new definition to cater for ALL angles.

New Definitions
y-axis
y
sin A =
y P(x,y) r
r
x
A cos A =
r
O o x x-axis
y
tan A =
x
Mar 23, 2013 www.mathsrevision.com
11

Trigonometry
S4 credit Example
Angles over 900
The radius line is 2cm. (1.2, 1.6)
The point (1.2, 1.6).

Find sin cos and tan for
the angle.
1.6 Check answer
sin 53o = = 0.80
2 with calculator
1.2
cos 53 =
o
= 0.60
2

tan 53 =
o 1.6
= 1.33 53o
23 Mar 2013 1.2 Created by Mr Lafferty Maths Dept

Trigonometry
S4 credit Example 1 Angles over 900
The radius line is 2cm.
Check answer
The point (-1.8, 0.8).
with calculator

Find sin cos and tan for
the angle.
(-1.8, 0.8)
0.8
sin127o = = 0.40
2
127o
−1.8
cos127 =o
= −0.90
2
0.8
tan127 =
o
= −0.44
23 Mar 2013 −1.8 Created by Mr Lafferty Maths Dept

Trigonometry Summary of
results
S4 credit Example All Quadrants
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 0o
180o
135o 315o
180o + xo 360o - xo
150o 330o
Tan +ve Cos +ve
Sin x Cos x Tan x

270o

What Goes In The Box ?
S4 credit

Write down the equivalent values of the following
in term of the first quadrant (between 0o and 90o):

1) Sin 135o sin 45 o 1) Sin 300o - sin 60o

2) Cos 150o -cos 45o 2) Cos 360o cos 0o

3) Tan 135 o -tan 45o 3) Tan 330 o - tan 30o

4) Sin 225 o -sin 45o 4) Sin 380 o sin 20o

-cos 90o - cos 80o
5) Cos 270o 5) Cos 460o

Trigonometry
S4 credit
Angles over 900

Now try MIA
Ch11 Ex3.1 Ch11
(page 222)


Starter
S4 Credit

1. True or false

2x +7x+6 = ( 2x + 3)(x + 2)
2

2. A TV is reduced by 20% to £200.
What was the original price.

Q3. Solve (2x-1)(x-1) = 0

Sine Graph
S4 Credit


1. To investigate graphs of 1. Identify the key points
the form for various graphs.

y = a sin xo

y = a cos xo

y = tan xo


Key Features

Sine Graph value at x = 90
Max
Zeros at 0, 180o and 360o
o

S4 Credit
Minimum value at x = 270o

Key Features
Domain is 0 to 360o
(repeats itself every 360o)
Maximum value of 1
Minimum value of -1


What effect
y = sinxo
does the number
at the front Sine Graph y = 2sinxo
have on the
S4 Credit graphs ? y = 3sinxo
3 y = 0.5sinxo
y = -sinxo

2

1

0
90o 180o 270o 360o
-1
What effect
-2 does the
negative sign
have on the
-3 graphs ? by Mr. Lafferty
created

Sine Graph
S4 Credit

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.

y = 5sinxo
Sine Graph
y = 4sinx o

y = sinxo
S4 Credit

6 y = -6sinxo

4

2

0
90o 180o 270o 360o
-2

-4

-6 created by Mr. Lafferty

Cosine Graphsat 90 and 270
Key Features
Zeros o o

Max value at x = 0o and 360o
S4 Credit
Minimum value at x = 180o

Key Features
Domain is 0 to 360o
Maximum value of 1
Minimum value of -1


What effect
y = cosxo
does the number
at the front Cosine 2cosx
y= o
have on the
S4 Credit graphs ? y = 3cosxo
3 y = 0.5cosxo
y = -cosxo

2

1

0
90o 180o 270o 360o
-1

-2


y = 2cosxo
Cosine Graph
y = 4cosx o

S4 Credit y = 6cosxo
6 y = 0.5cosxo
y = -cosxo

4

2

0
90o 180o 270o 360o
-2

-4


Key Features

Tangent Graphs Zeros at 0 and 180o

S4 Credit

Key Features

Domain is 0 to 180o


Tangent Graphs
S4 Credit


Tangent Graph
S4 Credit

y = a tan (x)


Period of a Function
S4 Credit

When a pattern repeats itself over and over,
it is said to be periodic.

Sine function has a period of 360o

Let’s investigate the function

y = sin bx

What effect
does the number
in front of x Sine Graph y = sinxo
have on the y = sin2xo
S4 Credit graphs ?
y = sin4xo
3
y = sin0.5xo

2

1

0
90o 180o 270o 360o
-1

-2


Trigonometry Graphs
S4 Credit

y = a sin (bx)

How many times
it repeats
itself in 360o


Cosinecosx
y= o

y = cos2xo
S4 Credit
y = cos3xo
3

2

1

0
90o 180o 270o 360o
-1

-2


Trigonometry Graphs
S4 Credit

y = a cos (bx)

How many times
it repeats
itself in 360o


Trigonometry Graphs
S4 Credit

y = a tan (bx)

How many times
it repeats
itself in 180o


Write down the
equations for the
graphs shown ? Trig Graph y = 0.5sin2xo
y = 2sin4xo
S4 Credit
Combinations y = -3sin0.5xo
3

2

1

0
90o 180o 270o 360o
-1

-2


Write down
y = 1.5cos2xo
equations for the
graphs shown? Cosine y = -2cos2xo
Combinations
S4 Credit y = 0.5cos4xo
3

2

1

0
90o 180o 270o 360o
-1

-2


Combination Graphs
S4 Credit

Now Try MIA Ch11
Ex 5.1
Page 227


C moves the graph
Trigonometry Graphs = 360
up or down in the
Period
o

y-axis direction b
S4 Credit

y = a sin (bx) + c

How many times
it repeats
a - Amplitude
itself in 360o


Sine Graph
S4 Credit
Simply move
graph up by 1

1

0
45o 90o 180o 270o 360o

Given the basic y = sin x
-1
graph what does the
graph of y = sin x + 1 look
like? created by Mr. Lafferty

Cosine Graph
Given the y = cos x graph.
What does the graph of
y = cos x – 0.5 look like?
S4 Credit
Simply move
down by 0.5

1

0
90o 160 180o
o
270o 360o

-1


Write down
equations for
graphs shown ? Trig Graph= 0.5sin2x
y o
+ 0.5

Combinationsy = 2sin4x - 1
o
S4 Credit

3

2

1

0
90o 180o 270o 360o
-1

-2


Write down
equations for the
graphs shown? Cosine y = cos2xo + 1
Combinationsy = -2cos2x - 1
o

S4 Credit

3

2

1

0
90o 180o 270o 360o
-1

-2


Combination Graphs
S4 Credit

Now try MIA
Ch11 Ex 5.2
Page 231


Starter
S4 Credit

1. Make b the subject of the formula

c=b+d

2. Use the quadratic formula to solve
x + 6x + 2
2

3. Sketch the function y = 2sin4x

S4 Credit


1. To explain how to solve 1. Use the rule for solving any
trig equations of the form ‘ normal ‘ equation
a sin xo + 1 = 0

2. Realise that there are many
solutions to trig equations
depending on domain.


S4 Credit

Sin +ve All +ve
180o - xo

180o + xo 360o - xo
Tan +ve Cos +ve
1 2 3 4


Solving Trig Equations what
Graphically
S4 Credit
a sin xo + b = 0 we trying to
are
solve
Example 1 :

Solving the equation sin xo = 0.5 in the range 0o to 360o

sin xo = (0.5)
xo = sin-1(0.5)
xo = 30o

There is another solution

xo = 150o
1 2 3 4
created by Mr. Lafferty (180o – 30o = 150o)

Graphically
S4 Credit
a sin xo + b = 0 we trying to
are
solve
Example 1 :

Solving the equation 3sin xo + 1= 0 in the range 0o to 360o

sin xo = -1/3

Calculate first Quad value
xo = 19.5o
x = 180o + 19.5o = 199.5o

1 2 3 4 ( 360o - 19.5o = 340.5o)

Graphically
S4 Credit
a cos xo + b are we trying to
=0
solve
Example 1 :

Solving the equation cos xo = 0.625 in the range 0o to 360o

cos xo = 0.625
xo = cos -1 0.625

xo = 51.3o
(360o - 53.1o = 308.7o)
1 2 3 4

Graphically
S4 Credit
a tan xo + b are we trying to
=0
solve
Example 1 :

Solving the equation tan xo = 2 in the range 0o to 360o

tan xo = 2
xo = tan -1(2)

xo = 63.4o
x = 180o + 63.4o = 243.4o
1 2 3 4

S4 Credit

Now try MIA Ch11
Ex6.1, 6.2 and 7.1

(page 236)


Starter
S4 Credit

1. Make a the subject of the formula

5 = 10b + a

2. Use the quadratic formula to solve
x + 5x + 1
2

3. Sketch the function y = 4sin3x

S4 Credit


1. To explain some special 1. Know and learn the two
trig relationships special trig relationships.

sin 2 xo + cos 2 xo = ?
2. Apply them to solve
and problems.

tan xo and sin x
cos x


S4 Credit

Lets investigate

sin 2xo + cos 2 xo = ?

Calculate value for x = 10, 20, 50, 250

sin 2xo + cos 2 xo = 1 Learn !


S4 Credit

Lets investigate

sin xo
tan xo and
cos xo
Calculate value for x = 10, 20, 50, 250

sin xo
tan xo = cos xo Learn !


S4 Credit

Now try MIA Ex8.1

Ch11 (page 238)


Tso math trig graphs

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