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Unit 4 Trigonometry

Allied Angles:

To connect the trigonometrical ratios of with those of U 

Thus

Similarly

sin (90 r  U) = cosU  cosec (90 r  U) = sec U 

cos (90 r  U) = sin U  sec (90 r  U) = cosec U 

tan (90 r  U) = cot  U  cot (90 r  U) = tan U 

To connect the trigonometrical ratios of 90 r +U wi th those U 

sin (90 r + U) = cosU cosec (90 r + U) = sec U 

cos (90 r + U) = sin U sec (90 r + U) = cosec U 

tan (90 r + U) = cot  U cot (90 r + U) =  tan U 

To express the trigonometrical ratios of 18 0 r  U in terms of those of U 

sin ( 18 0 r  U) = sin U cosec ( 18 0 r  U) = cosec U 

            

tantancoscos,sinsin !!! and 

UUUUUU cotcotsecsec,coscos !!! and ecec

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cos ( 18 0 r  U) = cosU sec ( 18 0 r  U) = sec U 

tan ( 18 0 r  U) =  tan U cot ( 18 0 r  U) = cot  U 

Multiple and Sub-multiple angle

The angles 2A, 3A, 4A etc., are called multiple angles. And etc., are called submultiple angles.

sin 2A = 2 sin A cos A 

cos 2A = cos2 A sin

2 A = 1  2 

sin2 A = 2 cos A

2  1 

tan 2A = 

Unit 5 Limits and continuitye

Theorem(): It (an) and (bn) and two sequences converging to a and b respectively, then

a)  (an + bn) a + b 

b)  (an bn) a b

A

A2

tan1

tan2

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c)  (kan) ka 

d)  (anbn) ab 

e) providedbn{ 0 for all n and b { 0. 

�  We study important properties of limits of functions as a theorem.We can evaluate limits using

this theorem (noted as a proposition).

�  Proposition: 

where k is a constant

Theorem: Let k be a constant, f and g functions having limit at a and n a positive integer. Then the

following hold good.

axIt ax

!p

k k It ax

!p

22axIt 

ax!

p

33axIt 

ax!

p

nn

axaxIt  !

p

axIt ax

!p

�  Two important trigonometric limits: 

If f(x) = sin x, then

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Rules of Integration

2 Sin A Cos B = Sin (A + B) + Sin (A B)  

2 Cos A Sin B = sin (A + B) Sin (A B)  

2 Cos A Cos B = Cos (A + B) Cos (A B)  

2 sin A Sin B = Cos (A B) Cos (A + B)  

More formulas in integration 

Using the method of substitution or integration by parts, we can derive the following formulas

Integral of functions of the form

xd x

xbxa

sincos

sincos

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Method 

Step 1: Let numerator = A (Denominator) + (Denominator)

Step 2: Find the values of A and B

Step 3: Split the function and integrate

We have already seen this

Completion of squares

All the subsequent methods use a technique called Completing the square. It is simply writing a

quadratic expression in the form, a2

+ x 2, 

a2

x 2, x 

2 a

2. 

�  Integration of functions of the form 

Method:

Step 1:Write

Step 2: Find the values of A and B

Step 3: Split the function and integrate

Integration of functions of the form

Method: Write in the form

Integration of functions of the form

cbxax

mIx

2

Bcbxaxdx

d §   

mIx ! 2

cbxax 2

1

cbxax 2 222222 xaor axor ax

cbxax

mIx

2

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Method: 

Step 1:Write

Step 2: Find the values of A and B

Step 3: Split the function and integrate

Integrals of functions of the form

Method:Write in form

and integrate

Integration of functions of the form

Method: 

Step 1:Write

Step 2: Find the values of A and B

Step 3: Split the function and integrate

Bcbxaxdx

d  ̈  

2Im

cbxax 2

1

cbxax 2 222222 axor xaor xa

cbxaxmIx 2

Bc

bxaxdx

d ©   

mIx !

2