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### Exact Values

National 5

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

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Graphs of the form y = a sin bxo + c

Solving Trig Equations

Special trig relationships

created by Mr. Lafferty

Learning Intention

Success Criteria

- Recognise basic triangles and exact values for sin, cos and tan 30o, 45o, 60o .

- To build on basic trigonometry values.

- Calculate exact values for problems.

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Created by Mr Lafferty Maths Dept

2

2

2

60º

60º

60º

2

Exact Values

National 5

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

30º

3

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1

This triangle will provide exact values for

sin, cos and tan 30º and 60º

National 5

Consider the square with sides 1 unit

45º

2

1

1

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45º

1

1

We are now in a position to calculate

exact values for sin, cos and tan of 45o

National 5

Now try Ex 2.1

Ch11 (page 220)

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Created by Mr Lafferty Maths Dept

National 5

Learning Intention

Success Criteria

- Find values of sine, cosine and tangent over the range 0o to 360o.

- Introduce definition of sine, cosine and tangent over 360o using triangles with the unity circle.

- 2. Recognise the symmetry and equal values for sine, cosine and tangent.

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Created by Mr. Lafferty Maths Dept.

y

x

P(x,y)

y

x

Angles Greater than 90o

National 5

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

Demo Sin

Demo Cos

Demo Tan

New Definitions

y-axis

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r

Ao

x-axis

O

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Angles over 900

Example

National 5

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

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53o

Created by Mr Lafferty Maths Dept

Angles over 900

Example 1

National 5

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)

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127o

Created by Mr Lafferty Maths Dept

Trigonometry

All Quadrants

Example

National 5

Calculate the ration for sin cos and tan

for the angle values below.

90o

30o

210o

45o

225o

Sin +ve

All +ve

60o

240o

xo

180o - xo

120o

300o

0o

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180o

135o

315o

180o + xo

360o - xo

150o

330o

Cos +ve

Tan +ve

270o

Created by Mr Lafferty Maths Dept

What Goes In The Box ?

National 5

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

- Sin 300o
- Cos 360o
- Tan 330o
- Sin 380o
- Cos 460o

- Sin 135o
- Cos 150o
- Tan 135o
- Sin 225o
- Cos 270o

- sin 60o

sin 45o

cos 0o

-cos 45o

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- tan 30o

-tan 45o

sin 20o

-sin 45o

- cos 80o

-cos 90o

Angles over 900

National 5

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

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Created by Mr Lafferty Maths Dept

National 5

Learning Intention

Success Criteria

- Identify the key points for various graphs.

- To investigate graphs of the form
- y = a sin xo
- y = a cos xo
- y = tan xo

www.mathsrevision.com

created by Mr. Lafferty

Sine Graph

Zeros at 0, 180o and 360o

Max value at x = 90o

National 5

Minimum value at x = 270o

Key Features

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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 = sinxo

y = 2sinxo

y = 3sinxo

y = 0.5sinxo

y = -sinxo

Sine Graph

National 5

3

2

1

0

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90o

180o

270o

360o

-1

What effect does the negative sign have on the graphs ?

-2

-3

Demo

created by Mr. Lafferty

National 5

y = a sin (x)

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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

y = 4sinxo

y = sinxo

y = -6sinxo

Sine Graph

National 5

6

4

2

0

www.mathsrevision.com

90o

180o

270o

360o

-2

-4

-6

created by Mr. Lafferty

Cosine Graphs

Zeros at 90o and 270o

Max value at x = 0o and 360o

National 5

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

National 5

3

2

1

0

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90o

180o

270o

360o

-1

-2

Demo

-3

created by Mr. Lafferty

y = 4cosxo

y = 6cosxo

y = 0.5cosxo

y = -cosxo

Cosine Graph

National 5

6

4

2

0

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90o

180o

270o

360o

-2

-4

-6

created by Mr. Lafferty

Tangent Graphs

Zeros at 0 and 180o

National 5

Key Features

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Domain is 0 to 180o

(repeats itself every 180o)

created by Mr. Lafferty

National 5

y = a tan (x)

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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

National 5

When a pattern repeats itself over and over,

it is said to be periodic.

Sine function has a period of 360o

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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

National 5

3

2

1

0

www.mathsrevision.com

90o

180o

270o

360o

-1

-2

-3

Demo

created by Mr. Lafferty

National 5

y = a sin (bx)

How many times

it repeats

itself in 360o

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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

y = cos2xo

y = cos3xo

Cosine

National 5

3

2

1

0

www.mathsrevision.com

90o

180o

270o

360o

-1

-2

Demo

-3

created by Mr. Lafferty

National 5

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

National 5

y = a tan (bx)

How many times

it repeats

itself in 180o

Demo

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

National 5

3

2

1

0

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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

National 5

3

2

1

0

www.mathsrevision.com

90o

180o

270o

360o

-1

-2

-3

created by Mr. Lafferty

National 5

Now Try MIA Ch11 Ex 5.1

Page 227

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created by Mr. Lafferty

C moves the graph up or down in the y-axis direction

Trigonometry Graphs

National 5

y = a sin (bx) + c

Demo

How many times

it repeats

itself in 360o

a - Amplitude

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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

Simply move graph up by 1

National 5

1

0

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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

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

National 5

1

0

160o

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90o

180o

270o

360o

-1

created by Mr. Lafferty

Write down equations for graphs shown ?

y = 0.5sin2xo + 0.5

y = 2sin4xo- 1

Trig Graph

Combinations

National 5

3

2

1

0

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

National 5

3

2

1

0

www.mathsrevision.com

90o

180o

270o

360o

-1

-2

-3

created by Mr. Lafferty

National 5

Now try MIA Ch11 Ex 5.2

Page 231

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created by Mr. Lafferty

National 5

Learning Intention

Success Criteria

- Use the rule for solving any ‘ normal ‘ equation
- Realise that there are many solutions to trig equations depending on domain.

- To explain how to solve
- trig equations of the form
- a sin xo + 1 = 0

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created by Mr. Lafferty

National 5

Sin +ve

All +ve

180o - xo

180o + xo

360o - xo

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Cos +ve

Tan +ve

1

2

3

4

created by Mr. Lafferty

Graphically what are we trying to solve

a sin xo + b = 0

National 5

Example 1 :

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

sin xo = (0.5)

xo = sin-1(0.5)

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xo = 30o

There is another solution

xo = 150o

1

2

3

4

(180o – 30o = 150o)

created by Mr. Lafferty

Graphically what are we trying to solve

a sin xo + b = 0

National 5

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

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x = 180o + 19.5o = 199.5o

There is another solution

1

2

3

4

( 360o - 19.5o = 340.5o)

created by Mr. Lafferty

Graphically what are we trying to solve

a cos xo + b = 0

National 5

Example 1 :

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

cos xo = 0.625

xo = cos -1 0.625

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xo = 51.3o

There is another solution

(360o - 53.1o = 308.7o)

1

2

3

4

created by Mr. Lafferty

Graphically what are we trying to solve

a tan xo + b = 0

National 5

Example 1 :

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

tan xo = 2

xo = tan -1(2)

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xo = 63.4o

There is another solution

x = 180o + 63.4o = 243.4o

1

2

3

4

created by Mr. Lafferty

National 5

Now try MIA Ch11

Ex6.1, 6.2 and 7.1

(page 236)

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created by Mr. Lafferty

National 5

Learning Intention

Success Criteria

- Know and learn the two special trig relationships.
- Apply them to solve problems.

- To explain some special trig relationships
- sin 2 xo +cos 2xo = ?
- and
- tan xo and sin x
- cos x

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created by Mr. Lafferty

National 5

Lets investigate

sin 2xo + cos 2 xo = ?

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

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Learn !

sin 2xo + cos 2 xo = 1

created by Mr. Lafferty

sin xo

cos xo

cos xo

Solving Trig Equations

National 5

Lets investigate

tan xo

and

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

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Learn !

tan xo

=

created by Mr. Lafferty

National 5

Now try MIA Ex8.1

Ch11 (page 238)

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created by Mr. Lafferty

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