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SIGNS AND GRAPHS OF TRIGONOMETRIC FUNCTIONS

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Introduction

sign_trigno_1As done in coordinate geometry, the coordinate axes divide the coordinate plane into four quadrants. They are named as I quadrant, II quadrant, III quadrant and IV quadrant.

Now we will learn about the pattern of trigonometric ratios:

We have six trigonometric ratios:

Sine (Sin), Cosine (Cos), Tangent (Tan), Cosecant (Cosec), Secant (Sec), Cotangent (Cot)

sign_trigno_2We will discuss about trigonometric function in detail using Unit Circle:

Consider a unit circle with centre at origin of the coordinate axes. Let P (a, b) be any point on the circle with ∠ AOP = x radian, i.e., length of arc AP = x

We define cos x = a and sin x = b   - (i)


In Δ OMP is a right triangle, by Pythagoras theorem we have,


OM2 + MP2 = OP2


a2 + b2 = 1 (from (i))


Thus, for every point on the unit circle, we have


a2 + b2 = 1 or cos2 x + sin2 x = 1


Since one complete revolution subtends an angle of 2π radian at the centre of the circle,


∠AOB =  Π2 , ∠AOC = Π and ∠AOD = 3Π2


All angles which are multiples of Π2 are called quadrantal angles.


The coordinates of the points A, B, C and D are respectively (1, 0), (0, 1),  (-1, 0) and (0, -1).


So, we have,

Cos 0° = 1      Sin 0° = 0

Cos Π2 = 0      Sin Π2 = 1


Cos Π = - 1     Sin Π = 0


Cos 3Π2 = 0     Sin 3Π2 = -1


Cos 2Π = 1      Sin 2Π = 0



sign_trigno_3Signs of Trigonometric Functions

Now we will discuss about the signs of trigonometric ratios in different quadrant.

I quadrant

I quadrant lies between the coordinates A (1, 0) and B (0, 1).

Also, in the first quadrant (0 < x< Π2) a and b are both positive.


So, in I quadrant – All trigonometric ratios are positive.


II quadrant

II quadrant lies between the coordinates B (0, 1) and C (-1, 0)

In II quadrant (Π2 < x < Π) a is negative and b is positive.


So, in II quadrant – Sine and Cosecant are positive rest all are negative.


III quadrant

III quadrant lies between the coordinates C (-1, 0) and D (0, -1)

In III quadrant (Π < x < 3Π2) a and b both are negative.


So, in III quadrant – Tangent and Cotangent are positive rest all are negative.


IV quadrant

IV quadrant lies between the coordinates D (0, -1) and A (1, 0)

In IV quadrant (3Π2< x < 2Π) a is positive and b is negative.


So, in IV quadrant – Cosine and Secant are positive rest all are negative.


Important Remark:

We have a simple aid to memorise the signs of trigonometric ratios in different quadrants.

ALL SCHOOL TO COLLEGE

I        II         III      IV
•    Letter ‘A’ states that all trigonometric ratios are positive in the I quadrant.
•    Letter ‘S’ states that only Sine and its reciprocal are positive in the II quadrant.
•    Letter ‘T’ states that only Tangent and its reciprocal are positive in the III quadrant.
•    Letter ‘C’ states that only Cosine and its reciprocal are positive in the IV quadrant.


Graphs of Trigonometric Functions

sign_trigno_4

The above table is based on the observations of different trigonometric ratios in respective quadrants.


From this, we can have the graphs of each trigonometric ratio mentioned in the above table.


1. Sin x sign_trigno_5


2. Cos x sign_trigno_6


3. Tan x sign_trigno_7


4. Cosec x  sign_trigno_8


5. Sec x sign_trigno_9


6. Cot x  sign_trigno_10




Now try it yourself!  Should you still need any help, click here to schedule live online session with e Tutor!



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