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Introduction

Every angle is represented by one position of a revolving ray OP of length r. The starting position for ray OP is taken along +X axis. The angles in trigonometry can be positive or negative and can have any magnitude.

To define sinθ, cosθ, tanθ, draw the OP ray for angle θ If (x, y) represents the co-ordinates of P for that position then(Refer the fig.):

 sinθ =

 cosθ =

  tanθ =

                       

Sign of T- ratios in four quadrants

  • If revolving ray lies in Quadrant 1, x & y are positive, hence all T -ratios are positive.
  • If revolving ray lies in Quadrant 2, x is negative & y is positive, hence only sin θ and cosecθ  are positive and cosθ, secθ, tanθ and cotθ are negative.
  • If revolving ray lies in Quadrant 3, x is negative & y is negative, hence only tanθ and cotθ  are positive rest cosθ , sinθ are negative.
  • If revolving ray lies in Quadrant 4, x is positive & y is negative, hence only cos θ and secθ are positive rest all are negative.

Positive and Negative Angles

sin (/2 - θ) = cos θ                                  sin (/2 + θ) = + cos θ

cos(/2 - θ) = sin θ                                   cos(/2 + θ) =  -  sin θ

tan (/2 - θ) = cot θ                                  tan (/2 + θ) = - cot θ                                                              

 

 sin ( - θ)  = + sin θ                                  sin ( + θ) = - sin θ

cos( - θ)  = - cos θ                                  cos ( + θ)  = - cos θ

tan ( - θ) = - tan θ                                   tan ( + θ) = + tan θ

 

 

sin (3/2 - θ) =- cos θ                                  sin (3/2 + θ) = - cos θ

cos(3/2 - θ) = -sin θ                                   cos(3/2 + θ) =  +  sin θ

tan (3/2 - θ) = +cot θ                                   tan (3/2 + θ) = - cot θ

 

sin (2 - θ)  = - sin θ                                  sin (2 + θ) =  sin θ

cos(2 - θ)  = + cos θ                                  cos (2 + θ)  =  cos θ

tan (2 - θ) = - tan θ                                   tan (2 + θ) =  tan θ

 

sin (2n + θ) = sin θ

cos (2n + θ) = cos θ

tan(2n +θ) = tan θ

 

 

sin (-θ) = - sin θ

cos(-θ) = +cos θ

tan (-θ) = - tan θ

Variations of sin θ, cos θ, tan θ                                                               

By definition we have, sin θ = and cos θ =    as |r| >= |x|,|y| max(sinθ) = 1 and min(sinθ) = -1 and similarly for max(cosθ) = 1 and min(cosθ) = -1. 

Proof:- Seeing the right angled triangle we can surely say:-

Ø     as - r  yr         =>           - 1 y/r  1            =>           -1sin θ l

Ø     as - r  x r        =>           -1 x/r 1              =>           -1cos θ 1

Ø     as tan θ  = y/x     =>           -< tan θ <        

 

In general we can say that: 

sin θ and cos θ can never be greater than 1 or less than -1.

sec θ and cosec θ can never be between -1 and +1.

tan θ and cot θ can take any positive or negative value.

 

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