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# TRIGONOMETRY

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## Marazieah Fatin

on 21 February 2014

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#### Transcript of TRIGONOMETRY

In trigonometry, mathematicians study the relationships between the sides and angles of triangles. Right triangles, which are triangles with one angle of 90 degrees, are a key area of study in this area of mathematics.
Pythagoras Theorem

SOH
Trigonometric Ratios
We can use the sine rule to find the size of an angle or length of a side.
Sine Rule
Cosine Rule
Two triangles are same but the only difference is size.
Similar Triangle
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.
Some angle properties of triangle
Is the triangle right-angle?
When to use them?
Application in real life
TRIGONOMETRY?
What is
Hypotenuse
Opposite
a relation among the three sides of a right triangle.
C
A
B
c = a + b
2
2
2
7
4
c = 7 + 4
c = 49 + 16
c = 65
c = 65
c = 8.06
2
2
2
2
2
c ?
A ratio that describes a relationship between a side and angle of a triangle
CAH
TOA
Opposite
Hypotenuse
Sin = Opp
Hyp
(θ)
Opposite
Tan = Opp
(θ)
Hypotenues
Hyp
(θ)
Finding sides:
Finding angles:
a = b = c
sin(a) sin(b) sin(c)
Formula:
6
a
80°
60°
a = 7
sin(80) sin(60)
7 x sin80 = sin60a
6.69 = a
sin60

7.96 = a
sin(a) = sin(b) = sin(c)
a b c
Formula:
a
75
°
10
8

sin(a) = sin(75)
8 10

8 x sin(75) = 10 sin(a)
7.73 = sin(a)
10
sin-1 0.773 = a
50.6 = a
°
it relates the lengths of the sides of a triangle to the cosine of one of its angles
Finding sides:
Finding angles:
a = b + c - 2bcCos(a)
Formula:
2
2
2
a = b + c – 2bc cos(A)

a = 22 + 28 – (2 × 22 × 28 × cos(97°))

a = 484 + 784 – (-150.143)

a = 1418.143

a = 37.7
2
2
2
2
cos(A) = b + c – a
2bc

cos(P°) = 5 + 8 – 7
2 × 5 × 8

cos(P°) = 25 + 64 - 49
80
cos(P ) = 40
80
cos(P°) = 0.5

P° = cos–1(0.5)

= 60°

2
2
2
Cos(A) = b + c - a
2bc
2
2
2
Formula:
2
2
2
2
2
2
it commonly used in finding the height of towers and mountains.
it is used in navigation to find the distance of the shore from a point in the sea.
It is used in oceanography in calculating the height of tides in oceans.
It is used in finding the distance between celestial bodies.
The sine and cosine functions are fundamental to the theory of periodic functions such as those that describe sound and light waves.
Architects use trigonometry to calculate structural load, roof slopes, ground surfaces and many other aspects, including sun shading and light angles
Does the question involve any angle?
Do you know a side and its opposite angle?
Use Trig. Ratios
(SOH, CAH, TOA)
Use Pythagoras Theorem
Use Sine Rule
Use Cosine rule
YES
NO
YES
NO
YES
NO
The person name Pythagoras was the one who discovered Pythagoras Theorem
It was discovered by mystical an American Indian chief
It was invented by the 15th Century German Mathematician
It was discovered by Howard Eves, an American mathematician known for geometry and history of mathematician
Done by group 1:
Halim
Majeedah
Fatin

Similar triangles have:

all their angles equal
corresponding sides have the same ratio
A
B
We know all the sides in Triangle A, and we know the side 6.4 in Triangle B
The 6.4 faces the angle marked with two arcs as does the side of length 8 in triangle A. So we can match 6.4 with 8, and so the ratio of sides in triangle B to triangle A is:
6.4 to 8
Now we know that the lengths of sides in triangle B are all 6.4/8 times the lengths of sides in triangle A.

a faces the angle with one arc as does the side of length 7 in triangle A.
a = (6.4/8) × 7 = 5.6

b faces the angle with three arcs as does the side of length 6 in triangle A.
b = (6.4/8) × 6 = 4.8
Exterior angles get their name because they lie on the outsides of triangles.
Exterior Angle Theorem
Some angle properties of triangle
Triangle Angle Sum Theorem
The diagram above illustrates the Triangle Angle Sum Theorem
The sum of the measures of the interior angles of a triangle is 180.
Example:
180 - 120 - 34
= 26
C = 26
°
°
°
°
Example:
180 - 42 - 30
= 108

M1 = 108
180 - 108
= 72
M2 = 72
°
°
°
°
°
°
°
°
°
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