Miraculous world of Numbers: 76-Basics of Trigonometry

Friday, December 27, 2013

76-Basics of Trigonometry - 04 Important key points

Click here for the previous basics of trigonometry.

Now we will study the next part of trigonometry.

Today we will study trigonometric ratios of 0°, 30°, 45°, and 60°.

1) Trigonometric ratios of an angle q = 0°

Here angle AOP is 0°.

According to the diagram, OP = r, and the coordinates of point P are (r, 0). So the x-coordinate of point P is r and the y-coordinate of point P is 0.

So, all the trigonometric ratios of q = 0° with

x = r,

y = 0,

r = r.

a) sin 0° = y/r sin 0° = 0/r sin 0° = 0	b) cos 0° = x/r cos 0° = r/r cos 0° = 1	c) tan 0° = y/x tan 0° = 0/r tan 0° = 0
d) csc 0° = r/y csc 0° = r/0 csc 0° = ∞	e) sec 0° = r/x sec 0° = r/r sec 0° = 1	f) cot 0° = x/y cot 0° = r/0 cot 0° = ∞

2) Trigonometric ratios of an angle q = 30°

Here angle AOP is 30°.

We know that the side opposite to 30° is half the hypotenuse and the side opposite to 60° is √3/ 2 times the hypotenuse.

So if hypotenuse OP = r, then AP = r/2 and OA = (√3 r)/ 2.
Here the coordinates of point P will be P ((√3 r)/ 2, r/2).

So, all the trigonometric ratios of q = 30° with

x = (√3 r)/ 2,

y = r/2,

r = r.

a) sin 30° = y/r sin 30° = (r/2)/r sin 30° = 1/2	b) cos 30° = x/r cos 30° = [(√3 r)/2]/r cos 30° = √3/2	c) tan 30° = y/x tan 30° = (r/2)/(√3 r)/2 tan 30° = 1/√3
d) csc 30° = r/y csc 30° = r/(r/2) csc 30° = 2	e) sec 30° = r/x sec 30° = r/[(√3 r)/2] sec 30° = 2/√3	f) cot 30° = x/y cot 30° = [(√3 r)/ 2]/(r/2) cot 30° = √3

3) Trigonometric ratios of an angle q = 45°

Here angle AOP is 45°.

We know that the side opposite 45° is 1/√2 times the hypotenuse.

So if hypotenuse OP = r, then AP = r/√2 and OA = r/√2.
Here the coordinates of point P will be P (r/√2, r/√2).

So, all the trigonometric ratios of q = 45° with

x = r/√2,

y = r/√2,

r = r.

a) sin 45° = y/r sin 45° = [r/√2]/r sin 45° = 1/√2	b) cos 45° = x/r cos 45° = [r/√2]/r cos 45° = 1/√2	c) tan 45° = y/x tan 45° = [r/√2]/ [r/√2] tan 45° = 1
d) csc 45° = r/y csc 45° = r/[r/√2] csc 45° = √2	e) sec 45° = r/x sec 45° = r/[r/√2] sec 45° = √2	f) cot 45° = x/y cot 45° = [r/√2]/ [r/√2] cot 45° = 1

4) Trigonometric ratios of an angle q = 60°

Here angle AOP is 60°.

We know that the side opposite 60° is √3/2 times the hypotenuse and the side opposite 30° is half times the hypotenuse.

So if hypotenuse OP = r, then AP = (√3 r)/2 and OA = r/2.
Here the coordinates of point P will be P (r/2, (√3 r)/2).

So, all the trigonometric ratios of q = 60° with

x = r/2,

y = (√3 r)/2,

r = r.

a) sin 60° = y/r sin 60° = [(√3 r)/2]/r sin 60° = √3/2	b) cos 60° = x/r cos 60° = [r/2]/r cos 60° = 1/2	c) tan 60° = y/x tan 60° = [(√3 r)/2]/[r/2] tan 60° = √3
d) csc 60° = r/y csc 60° = r/[(√3 r)/2] csc 60° = 2/√3	e) sec 60° = r/x sec 60° = r/[r/2] sec 60° = 2	f) cot 60° = x/y cot 60° = [r/2]/[(√3 r)/2] cot 60° = 1/√3