trigonometry-4 ratios of the sides of triangles. the meaning of sin, cos and tan sin is short for...
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Trigonometry-4
Ratios of the Sides of Triangles.
The meaning of sin, cos and tansin is short for sine
cos is short for cosine
tan is short for tangent
Trigonometry
Triangles – Ratios of Sides
Construct a Right Angled Triangle
Ratios in a Right Angled Triangle
Triangles – Ratios of Sides
In triangle ABC, AB = 5 units, BC = 3 units and AC = 4 units.
A ratio is one number divided by another number.
BC
A
B
C
3
4
5
The ratio of side BC to side AC is written as: AC 4
3= 0,7
5=
Triangles – Ratios of Sides
In triangle ABC you are given the length of the sides.
A ratio is one number divided by another number.
BC
A
B
C
3
4
5
The ratio of side BC to side AB is written as: AB 5
3= 0,6=
Triangles – Ratios of Sides
In triangle ABC you are given the length of the sides.
A ratio is one number divided by another number.
AC
A
B
C
3
4
5
The ratio of side AC to side AB is written as: AB 5
4= 0,8=
Construct a Right Angled Triangle
The tools you need:
rulerpencil
protractor
set square
Construct a Right Angled Triangle with an angle of 30
1: Draw a line
Construct a Right Angled Triangle
1: Draw a line
Construct a Right Angled Triangle
2: Make a point on the line
Construct a Right Angled Triangle
3: Move the protractor onto the point
Construct a Right Angled Triangle
4: Mark your angle with a point
Construct a Right Angled Triangle
4: Mark your angle with a point
Construct a Right Angled Triangle
5: Draw a line through the two points
Construct a Right Angled Triangle
5: Draw a line through the two points
Construct a Right Angled Triangle
6: Now draw a right angle to complete the triangle.
Construct a Right Angled Triangle
6: Now draw a right angle to complete the triangle.
Construct a Right Angled Triangle
30
7: Write in the size of the angle.
Ratios in a Right Angled Triangle
Label the hypotenuse H, the opposite side O and the adjacent side A.
30
HO
A
Ratios in a Right Angled Triangle
Measure the length of H, O and A.
30
HO
A
Do you think everyone will get the
same answers?
No – all the triangles are different.
The only thing that is the same is the
angle.
Ratios in a Right Angled Triangle
Measure the length of H, O and A.
30
HO
A
A =
H =
O =
5,0
Remember – your answers won’t be the same.
Ratios in a Right Angled Triangle
Measure the length of H, O and A.
30
HO
A
A =
H =
O =
5,05,7
Remember – your answers won’t be the same.
Ratios in a Right Angled Triangle
Measure the length of H, O and A.
30
HO
A
A =
H =
O =
5,05,72,8
Remember – your answers won’t be the same.
Ratios in a Right Angled Triangle
30
HO
A
A = 5,0H = 5,7O = 2,8
OA
= 2,85,0
Work out the ratio: OA
=0.56
OA
=0.56
Ratios in a Right Angled Triangle
30
HO
A
A = 5,0H = 5.7O = 2,8
OH
= 2,85,7
Work out the ratio: OH
=0.49
OH
=0.49
OA
=0.56
Ratios in a Right Angled Triangle
30
HO
A
A = 5,0H = 5.7O = 2,8
AH
= 5.05,7
Work out the ratio: AH
=0.87
AH
=0.87
OH
=0.49
OA
=0.56
Ratios in a Right Angled Triangle
A = 5,0H = 5.7O = 2,8
Our Answers
OA
=0.56
OH
=0.49
AH
=0.87
Do you think our diagrams are very
accurate?
Probably not!With our tools we can’t be exact!
What should we do to our answers?
Ratios in a Right Angled Triangle
A = 5,0H = 5.7O = 2,8
Our Answers
OA
=0.56
OH
=0.49
AH
=0.87
The second decimal place won’t be
accurate.
Let’s round the answers to one decimal place.
Ratios in a Right Angled Triangle
A = 5,0H = 5.7O = 2,8
Our Answers
OA
=0.56
OH
=0.49
AH
=0.87
How do we round the answers to one
decimal place?
If the second decimal is 4 or less, then the first decimal stays the same.
If the second decimal is 5 or more, then the first decimal becomes bigger by 1.
OA
=0.6
Ratios in a Right Angled Triangle
A = 5,0H = 5.7O = 2,8
Our Answers
OH
=0.49
AH
=0.87
How do we round the answers to one
decimal place?
If the second decimal is 5 or more, then the first decimal becomes bigger by 1.
If the second decimal is 4 or less, then the first decimal stays the same.
OA
=0.6
Ratios in a Right Angled Triangle
A = 5,0H = 5.7O = 2,8
Our Answers
OH
=0.5
AH
=0.87
How do we round the answers to one
decimal place?
If the second decimal is 5 or more, then the first decimal becomes bigger by 1.
If the second decimal is 4 or less, then the first decimal stays the same.
OA
=0.6
Ratios in a Right Angled Triangle
A = 5,0H = 5.7O = 2,8
Our Answers
OH
=0.5
AH
=0.9
How do we round the answers to one
decimal place?
If the second decimal is 5 or more, then the first decimal becomes bigger by 1.
If the second decimal is 4 or less, then the first decimal stays the same.
OA
=0.6
Ratios in a Right Angled Triangle
A = 5,0H = 5.7O = 2,8
Our Answers
OH
=0.5
AH
=0.9
Does everyone in the class get the same answers?
YES ! Amazing!
From all the different sized triangles you drew we get the same answers for the ratios.
OA
=0.6
Ratios in a Right Angled Triangle
A = 5,0H = 5.7O = 2,8
Our Answers
OH
=0.5
AH
=0.9
On a scientific calculator do the
following:
Enter 30.
Press the ‘tan’ button.
Round the answer to 1 decimal.
Answer is same as:
OA
=0.6
Ratios in a Right Angled Triangle
A = 5,0H = 5.7O = 2,8
Our Answers
OH
=0.5
AH
=0.9
On a scientific calculator do the
following:
Enter 30.
Press the ‘sin’ button.
Round the answer to 1 decimal.
Answer is same as:
OA
=0.6
Ratios in a Right Angled Triangle
A = 5,0H = 5.7O = 2,8
Our Answers
OH
=0.5
AH
=0.9
On a scientific calculator do the
following:
Enter 30.
Press the ‘cos’ button.
Round the answer to 1 decimal.
Answer is same as:
OA
=0.6
Ratios in a Right Angled Triangle
A = 5,0H = 5.7O = 2,8
Our Answers
OH
=0.5
AH
=0.9
Answers on a scientific calculator:
Gives same answer as ‘tan’ button.Gives same answer as ‘sin’ button.Gives same answer as ‘cos’ button.
Ratios in a Right Angled Triangle
Instead of an angle of 30 in the triangle, you can put in any angle
and get the same results.
Use the ‘tan’ button.
Use the ‘sin’ button.
Use the ‘cos’ button.
OA
Ratio:
OH
Ratio:
AH
Ratio:
Ratios in a Right Angled Triangle
Instead of drawing triangles and measuring sides to get the ratios, we can just use a scientific calculator.
In any right angled triangle:
O
A
H
OA
Tan =
OH
Sin =
AH
Cos =
Ratios in a Right Angled Triangle
Tan 20 = 0,36
OAOHAH
O
20A
HO
40A
HO
60A
H
Tan 40 = 0,83
Tan 60 = 1,73
sin 20 = 0,34 sin 40 = 0,64 sin 60 = 0,86
cos 20 = 0,93
cos 40 = 0,76
cos 60 = 0,5
These are true for any triangle with these
angles.
Ratios in a Right Angled Triangle
We have to remember these trigonometric ratios.
In any right angled triangle:
O
A
H
OA
Tan =
OH
Sin =
AH
Cos =
Some Old Hens
Cackle And Howl
Till Old Age
Ratios in a Right Angled Triangle
O
A
H
OA
Tan =
OH
Sin = AH
Cos =
Some Old HensCackle And Howl
Till Old Age
Well Done!