Saturday, December 28, 2013

Basics of Trigonometry - 05

Blog-86

Today we will study trigonometric ratios of 90°, 180°, and 270°.

5) Trigonometric ratios of an angle O = 90°
Here angle AOP is of 90°.
According to the diagram, OP = r and the coordinates of point P are (0, r). So x-coordinate of point P is 0 and the y-coordinate of point P is r.
So, all the trigonometric ratios of O = 90° with
 x = 0, y = r, r = r.
a) sin 90° = y/r
    sin 90°  = r/r   
    sin 90°  = 1
b) cos 90° = x/r
     cos 90° = 0/r
     cos 90° = 0
c) tan 90° = y/x
    tan 90°  = r/0  
    tan 90°  = ∞
d) csc 90° = r/y
    csc 90° =  r/r
    csc 90° =  1
e) sec 90° = r/x
    sec 90° = r/0          sec 90° = ∞
f) cot 90° = x/y
    cot 90° = 0/r   
    cot 90° = 0


6) Trigonometric ratios of an angle O = 180°
Here angle AOP is of 180°.
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 O = 180° with
 x = -r, y = 0, r = r.
 
a) sin 180° = y/r
    sin 180°  = 0/r        sin 180°  = 0
b) cos 180° = x/r
     cos 180° = -r/r
     cos 180° = -1
c) tan 180° = y/x
     tan 180° = 0/-r        tan 180° = 0
d) csc 180° = r/y
    csc 180° =  r/0
    csc 180° =  ∞
e) sec 180° = r/x
    sec 180° = r/-r              sec 180° = -1
f) cot 180° = x/y
    cot 180° = -r/0 
    cot 180° = -∞

7) Trigonometric ratios of an angle O = 270°
Here angle AOP is of 270°.
According to the diagram, OP = r and the coordinates of point P are (0, -r). So x-coordinate of point P is 0 and the y-coordinate of point P is -r.

So, all the trigonometric ratios of O = 270° with
 x = 0, y = -r, r = r.

a) sin 270° = y/r
    sin 270° = -r/r   
    sin 270° = -1
b) cos 270° = x/r
     cos 270° = 0/r
     cos 270° = 0
c) tan 270° = y/x
    tan 270° = -r/0  
    tan 270° = ∞
d) csc 270° = r/y
    csc 270° =  r/-r
    csc 270° =  -1
e) sec 270° = r/x
    sec 270° = r/0
    sec 270° = ∞
f) cot 270° = x/y
   cot 270° = 0/-r   
   cot 270° = 0

Actually, it is very easy to understand the concept once we found out the values of x, y and r for the point P on the terminal ray.

In the next Blog, we will study all the trigonometric ratios for the following groups.
Group-01: 120° (90° + 30°), 210° (180° + 30°), 300° (270° + 30°).
Group-02: 135° (90° + 45°), 225° (180° + 45°), 315° (270° + 45°).
Group-03: 150° (90° + 60°), 240° (180° + 60°), 330° (270° + 60°).

Anil Satpute