Friday, December 27, 2013

Basics of Trigonometry - 04

Blog-85
Dear Students,
Now we will study the next part of trigonometry. 

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

1) Trigonometric ratios of an angle =  0°

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

So, all the trigonometric ratios of 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 30°
Here angle AOP is of 30°.

We know that the side opposite to 30° is half the hypotenuse and 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 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 45°
Here angle AOP is of 45°.

We know that the side opposite to 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/√2r/√2).
So, all the trigonometric ratios of 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 60°
Here angle AOP is of 60°.

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

In the next blog, we will study the trigonometric ratios of 90°, 180° and the tabulated form of all the trigonometric ratios.

Anil Satpute