Calculus, Derivative and anti-derivative of logarithmic and trigonometric functions

Calculus
shahbaz ahmed
January 2024

OBJECTIVES OF THE DOCUMENTS Document is written to combine key results in
trigonometry ,derivative and ant-derivative(integral) and such co-related concepts and skills as a
whole in a single tutorial. It includes:

• Proofs of some important trigonometric functions having wide use in Calculus
• Derivative of trigonometric and logarithmic functions.
• Anti-derivative (integral) of trigonometric and logarithmic functions


Introduction to some important trigonometric formulas




Reference to the figure

1

, Triangle OEC is a right angle triangle with m̸ OEC = 90◦ .Similarly triangle OCD is a right

angle triangle with m̸ OCD = 90◦ .

Also m̸ EOC = α

m̸ OCB = α

m̸ BCD = 90 − α

m̸ BDC = α

m̸ COD = β

Also triangle OAD is a right angle triangle with m̸ OAD = 90◦

m̸ AOD = α + β

Now in the right angle triangle OAD
AD
sin (α + β) = OD
AB+BD
sin (α + β) = OD

Since AB = EC
EC+BD
sin (α + β) = OD
EC BD
sin (α + β) = OD + OD
EC OC BD DC
sin (α + β) = OC OD + DC OD




2

, sin (α + β) = sin α cos β + cos α sin β ............ eq (1)


3