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**

Its all about Triangle..

Hypotenuse

Opposite

Adjacent

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

Adjacent

Tan = Opp

Adj

(θ)

Adjacent

Hypotenues

Cos = Adj

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

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

°

°

°

°

°

°

°

°

°