Tuesday, January 7, 2014

Basics of Trigonometry - 07

Blog-88
Dear Students,
Now we will study the next part of trigonometry. 


Today we will study trigonometric ratios of three groups as shown bellow.
Group-02: 135° (90° + 45°)225° (180° + 45°)315° (270° + 45°).

2) Trigonometric ratios of Group-02: 135° (90° + 45°)225° (180° + 45°)315° (270° + 45°).

a) An angle 135°
Here angle XOP is of 135° (Anti-clock-wise-direction).

Here, inclination of ray OP is 135°, so angle AOP 45° and angle OPA is 45°

We know that the side opposite to 45° is 1/√2 times the hypotenuse.
So if hypotenuse OP = r, then  AP = OA = r/√2 (side opposite of 45°).

As point A is to the negative side of X-axis, x-coordinate of point A will be - r/√2. In the same way point P is in the 2nd quadrant, so y-coordinate of point P will be r/√2. So, the coordinates of point P will be (-r/√2r/√2).

So, all the trigonometric ratios of 135° with
 x = r/√2,
 y = r/√2,
 r = r.
a) sin 135° = y/r
    sin 135° = (r/√2)/r   
    sin 135° = 1/√2
b) cos 135° = x/r
    cos 135° = (- r/√2)/r        cos 135° = - 1/√2
c) tan 135° = y/x
    tan 135° = (r/√2)/(- r/√2)                      tan 135° = - 1
d) csc 135° = r/y
    csc 135° =  r/(r/√2)     
    csc 135° = √2
e) sec 135° = r/x
    sec 135° = r/(- r/√2)      
    sec 135° = - √2
f) cot 135° = x/y
    cot 135° = (- r/√2)/(r/√2)   
    cot 135° = - 1


b) An angle 225°
Here angle XOP is of 225° (Anti-clock-wise-direction).

Here, inclination of ray OP is 225°, so angle AOP 45° and angle OPA is 45°

We know that the side opposite to 45° is 1/√2 times the hypotenuse.
So if hypotenuse OP = r, then  AP = OA = r/√2 (side opposite of 45°).

As point A is to the negative side of X-axis, x-coordinate of point A will be - r/√2. In the same way point P is in the 3rd quadrant, y-coordinate of point P will be - r/√2. So, the coordinates of point P will be (-r/√2, -r/√2)

So, all the trigonometric ratios of 225° with
 x = r/√2,
 y = - r/√2,
 r =  r.
a) sin 225° = y/r
    sin 225° = (r/√2)/r    sin 225° = -1/√2
b) cos 225° = x/r
    cos 225° = (-r/√2)/r        cos 225° = -1/√2
c) tan 225° = y/x
    tan 225° = (r/√2)/(-r/√2)                  tan 225° =  1
d) csc 225° = r/y
    csc 225° =  r/(r/√2)
    csc 225° = -√2
e) sec 225° = r/x
    sec 225° = r/(r/√2)      
    sec 225° = -√2
f) cot 225° = x/y
    cot 225° = (r/√2)/(r/√2)
    cot 225° = 1

c) An angle 315°
Here angle XOP is of 315° (Anti-clock-wise-direction).

Here, inclination of ray OP is 315°, so angle AOP 45° and angle OPA is 45°


We know that the side opposite to 45° is 1/√2 times the hypotenuse.
So if hypotenuse OP = r, then  AP = OA = r/√2 (side opposite of 45°).

As point A is to the positive side of X-axis, x-coordinate of point A will be r/√2. In the same way point P is in the 4th quadrant, so y-coordinate of point P will be - r/√2. So, the coordinates of point P will be (r/√2, -r/√2)

So, all the trigonometric ratios of 315° with
 x = r/√2,
 y = - r/√2,
 r =  r.


a) sin 315° = y/r
    sin 315° = (-r/√2)/r   
    sin 315° = -1/√2
b) cos 315° = x/r
    cos 315(r/√2) r        cos 315° = 1/√2
c) tan 315° = y/x
    tan 315° = (-r/√2)/(r/√2)
     tan 315° =  -1
d) csc 315° = r/y
    csc 315° =  r/(-r/√2) 
   csc 315° = -√2
e) sec 315° = r/x
    sec 315° = r/(r/√2)      
    sec 315° = √2
f) cot 315° = x/y
    cot 315° = (r/√2)/(-r/√2)
    cot 315° = -1

Group-03: 150° (90° + 60°)240° (180° + 60°)330° (270° + 60°) will be published in the next Blog.