"A hundred years from now, it will not matter what kind of car I drove, what kind of house I lived in, how much money I had in the bank...but the world may be a better place because I made a difference in the life of a child." -- Forest Witcraft

"Education would be much more effective if its purpose was to ensure that by the time they leave school every boy and girl should know how much they do not know and be imbued with a lifelong desire to know it." -- William Haley

"One looks back with appreciation to the brilliant teachers, but with gratitude to those who touched our human feelings. The curriculum is so much necessary material, but warmth is the vital element for the growing plant and for the soul of the child." -- Carl Jung

There are two good reasons to be a teacher – June and July.

"We spend the first twelve months of our children's lives teaching them to walk and talk, and the next twelve years telling them to sit down and shut up."

"A statistician can have his head in an oven and his feet in ice, and he will say that on the average he feels fine."

• I have heard that parallel lines do meet, but they are very discrete

BY:Mrs.ROOHI JILANI

Measuring Angles : In Degrees or Radians

The angle, , can be measured in degrees. This represents the turn required to move from one line to the other in the direction shown.

This turn is measured in degrees. Degrees are a unit measuring turning where 360o

is a full turn.

360o

If we imagine a circle of radius 1 unit, then a full turn would be a full circle and the point A moves would be the same as the circumference of the circle

Radians is another measure for angles. This time you represent the angle as the distance point A moves around the circumference of an imaginary circle.

A

360o = 2 radians (or 2 c)

1o = 2 c

360o

1 c = 360o

2

r

r

Length of arc, LL = (2 r) 360o

Area of sector, AA = (r 2) 360o

L = (2 r) = r 2

A = (r2) = ½ r2 2

In degrees …In radians …

Here we have a sector draw with angle . This sector has an arc length of L and an area of A.

L

Area,A

Uses of radians

1. Convert from degrees to radians1. 30o

2. 145o

3. 500o

4. -60o

2. Convert from radians to degrees1. 2/3 rads

2. 7/5 rads

3. -5/8 rads

4. 0.5 rads

Calculate length of the arc and areas for these sectors,

a)

b)

c)

Radius = 4cm

Radius = 6.3cm

Radius = 14cm

Note : angles in radians

Adjacent

Opp

osite

Hypotenuse

sine = opposite ÷ hypotenuse

cosine = adjacent ÷ hypotenuse

tangent = opposite ÷ adjacent

0o

90o

45o

60o

60o

60o

30o

Hypotenuse = Adjacent

Opposite = 0

sin 0o = 0 cos 0o = 1 tan 0o = 0

Hypotenuse = Opposite

Adjacent = 0

sin 90o = 1 cos 90o = 0 tan 90o = undefined

Adjacent = Opposite = xHypotenuse = x2sin 45o = 1/2 cos 45o = 1/2 tan 45o = 1

x

x

x

x

xFor 60o Hypotenuse = xAdjacent = x Opposite = x 3

2 2 sin 60o = 3 cos 60o = 1 tan 60o = 3 2 2

For 30o Hypotenuse = xOpposite = x Adjacent = x 3

2 2 sin 30o = 1 cos 30o = 3 tan 30o = 1

2 2 3

Some Standard Solutions …

(deg)

0 30 45 60 90

(rads)

0 6

4

3

2

sin 0 ½ 1

2

3

21

cos 1 3

2

1

2½ 0

tan 0 1

31 3 -

0o180o

90o

270o

0o180o

90o

270o

0o180o

90o

270o

0o180o

90o

270o

Opposite = +

Adjacent = +Opposite = +

Adjacent = -

Opposite = -

Adjacent = -

Opposite = -

Adjacent = +

+

+

+

+

+

+

- -

--

--

Images from BBC AS Guru

Sin + All +

Tan + Cos +

CAST Diagram

Sin + All +

Tan + Cos +

Solve : sin x = 0.5 for the range 0 ≤ x ≤ 360o

x = arcsin 0.5 = 30o but sin is positive in two quadrants so x = 30o or (180 – 30)=150o

Solve : sin x = 0.5 for the range 0 ≤ x ≤ 360o

x = arcsin 0.5 = 30o but sin is positive in two quadrants so x = 30o or (180 – 30)=150o

Find all the angles (in degrees) in the given range1. Sin x = - ½ for the range 0 ≤ x ≤ 360o

2. Cos 2x = for the range -360o ≤ x ≤ 360o

3. Tan (2x+40o) = for the range -180o ≤ x ≤ 180o

Find all the angles (in radian) in the given range1. Sin x = for the range 0 ≤ x ≤

2. Cos 2x = -for the range -2 ≤ x ≤ 2

3. Tan (2x ½ = for the range - ≤ x ≤

A physicist and an engineer are in a hot-air balloon. Soon, they find themselves lost in a canyon somewhere. They yell out for help: "Helllloooooo! Where are we?" 15 minutes later, they hear an echoing voice: "Helllloooooo! You're in a hot-air balloon!!" The physicist says, "That must have been a mathematician." The engineer asks, "Why do you say that?" The physicist replied: "The answer was absolutely correct, and it was utterly useless."