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Trigonometric Ratios Re-Defined

Consider a unit circle with centre O and a revolving line OP. Let OP revolve in an anticlockwise direction and arrive at a point (a, b) in the 1st quadrant (a >0 and b> 0).
PM is drawn perpendicular to OA

Trigonometric ratios (circular functions) are defined as follows:



We notice all the ratios are positive, since a and b are positive.

Note:- From the definition, observe that tan θ and sec θ and not defined if
a = 0 and cot θ and cosec θ are not defined if b =0.
Observation: not defined.

Move to II quadrant now
The angle rotated through is π θ .
PM = b and OM = a

Definitions:-


Here we notice that sin and cosec functions remain the same and the other ratios change sign. (because negative in the second quadrant)

Observation:- When b = 0, angle is π

so we notice: sin π = 0, cos π = 1, tan π = 0, cosec π , cot π not defined

and sec π = 1.

Table II

Signs of Trigonometry Ratios
Quadrant sinθ cosθ tanθ cosecθ secθ cotθ
I + + + + + +
II + - - + - -
III - - + - - +
IV - + - - + -

The same idea can be briefly represented like this

Table III
Values of Trigonometric Ratios for 0 2π
0
0 1 0 -1 0
1 0 -1 0 1
0 1 0 0

Note:- Any positive number divided by 0 is considered to be ∞ and negative number divided by 0 is -∞.

Important Results
  1. For all values of θ , cos( θ ) = cosθ and


This is because ( θ ) is an angle in the 4th quadrant (if θ < 90°) then cosine is positive, sine and tangent are negative.

This property leads to the conclusion:
Cosine is an even function
Sine is an odd function
Tangent is an odd function
The reciprocals of sin, cos and tan have the same property as the original functions

  1. Values of sin and cos functions repeat after one complete revolution. So we say that sin and cos have a period of 2π
Eg:

The Period of tan function is π .
Eg:

Values of sin and cos functions fluctuate between -1 and +1 where as tan values change from -∞ to ∞

Table IV Values of T - Ratios in the four quadrants
Quadrant sin x cos x tan x cosec x sec x cot x

I

0 to 1 1 to 0 0 to ∞ ∞ to 1 1 to ∞ ∞ to 0

II

1 to 0 0 to -1 -∞ to 0 1 to ∞ -∞ to -1 0 to -∞

III

0 to -1 -1 to 0 0 to ∞ -∞ to -1 -1 to -∞ ∞ to 0

IV

-1 to 0 0 to 1 -∞ to 0 -1 to -∞ ∞ to 1 0 to -∞
Working rule for finding trigonometric ratios of any angle (like etc where θ is acute angle)

Rule:-
  1. Write the angle in the form
  2. Determine the quadrant in which the terminal side of the angle lies.
  3. Determine the sign of the given Trigonometric ratio in that particular quadrant (Table II)
  4. If k is even, T - ratio of the given angle is the same as what is given in the question.
  5. If k is odd, change the function to its complementary ratio.
Use the above rules and complete the following table

Table V, θ is acute

Example 1:
Use the rules given above to find sin 2460°


Example 2:
1. Evaluate

Solution:-


Example 3:
Find the value of other trigonometric ratios if


Solution:-


[II quadrant only sinx & cosecx are positive]



[III quadrant only tanx & cotx are positive]

Example 4:
If,evaluate.

Solution:




Example 5:


Solution:
 
 




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