Calculus, Derivative and anti-derivative of logarithmic and trigonometric functions
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Course
Punjab University
Institution
Punjab University
Documents consists of proof of key formulas to be used in the study of Calculus. It contains:
(1) Proof of some important limits
(2) Derivative and integration of important formulas in trigonometric and logarithmic functions
(3) Linked concepts on derivative and anti-derivative (integral)
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
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