Sr no: 000 class: XI sub: Maths chap: 3 TRIGNOMETRY lesson: 01
THE BASIS OF TRIGNOMETRY:
All of us have used a protractor to make an angle. To make it we mark a point on a straight line (X-axis) with a pen. Then place the center of the protractor on this point. Go to the scale showing degrees and mark a point at say 3 deg. Join this point with the center chosen. This angle is 3 deg. To make bigger angle, say 10 deg we move counter clockwise and repeat the same method.
If you count all the number of degrees marked on protractor it is 180. But protractor is only HALF of circle.
If we place this protractor up-side down we obtain a full circle that now has 180=180=360 deg of angle.
This circle has some radius r units. Radius or size of the protractor does NOT matter.
- We must mark the second point on protractor, first being x-axis as 0 deg.
- We move along the CIRCUMFERENCE of the circle made by protractor. Circum = 2πr units ( we henceforth take all units of length as meters)
- Total angle on two protractors making a full circle is 360 deg at origin (0,0) of x-y axis.
- We can drop a perpendicular from a point A on the circumference to the X-axis like AP.
We get a right-angle triangle OPA.
- We measure angle from x-axis moving in anticlockwise direction and call this + angle.
- Measure of angle starts from 0 to 1,7,79,267,359 and back to 360=0 deg. Values of angle increase from 0 to 360 and again start from 0.
Thus angle 369 = 360 + 9 = 9 deg Thus adding 360 does not change value of angle.
Unless we connected the two points on protractor angle was not made. And we know we travelled along the circum of circle. Thus angle made at center is small or big depending on how much distance we travel along the circumference, which is NOT a straight line.
We know cirum = 2 x π x radius. And that full circle circumference SUBTENDS angle of 360 deg at center.
2 π radius means 3.28rads.
Similarly, if 360 deg = 2π, then one deg = (2π/360) rad and so on. And vice-versa.
We have this conversion formula now.
**NOTE that size of the circle does NOT matter. If we are given length of arc in meters, we can ask for value of radius and concert all circumference values to ONE RAD units.
Note the right-angle triangle that we made. We like right-angle because it has special properties and follows Pythagoras theorem.
Here too like a circle a right-angle is universally describable by two sides only. Or one side and one angle.
**How can we make use of this Right Angle triangle:
How many things we can do: For Our Right angle triangle defined in Gig #2
- B/D : let us name this ratio as cosine ϴ between lines B and D.
- H/D let us name this ratio as sine ϴ between lines B and D.
- B/H let us name this ratio as tangent ϴ between lines B and D.
Note that this angle FACES height H
- Angle B = 90 – angle A
- NOTE the arrow placed by us on D near vertex B. This will come later.
- When you select an angle then line facing the angle has becomes the H.
- You can rotate the right angle triangle in any manner but once you select an angle then side facing this angle is H.
** We divide x-y plane into 4 equal parts called quadrants as in Fig 1. When we complete a circle we cover all the four equal quadrants of cartesian coord.
Circumference of this circle is 2πr
To mark the beginning we start at X-axis.= that is x = radius, y = 0.
Each of the four parts of the cartesian coord is called a quadrant (1/4 of whole chart), These are numbered as I,II,III,IV quads in anti clock wise direction. Thus
QI= 0-90 dg
QII= 90 -180 de
QIII = 180-270
QIV= 270-360 deg.
Thus an angle of 0, or 360, or 720 or 1080 are all equal to 0 degree angle. Anything that happens is when angle is between 0 – 30s deg only.
**Because all properties are applicable to value os angles between 0 -90 only. Thus we can simplify an angle as:
91 deg = 90 + 1
179 deg = 90 + 89
276 eg = 270 + 6
359 deg = 270 + 89
Remember when you are in any of the four quad your behavior becomes DIFFERENT.
This is all that we need to learn TRIGNOMETRY.