Summary
Compound angle formulas are:
Half angle formulas are:
Function to trigonometric form:
In Fig 1, and are acute angles and
As
Hence,
Replacing B by -B in i)
Also from the diagram:
Hence,
Also:
We have derived the compound angle formulae above.
All the compound angle formulas are listed below:
Double Angle formulae
We use compound angle formulas from above to find double angle formula i.e
If:
Replacing B by A:
Similar replace B by A:
Finally,
Example #1
Q. Given that , where is obtuse, find the exact value of
i) sin 2
ii) cos 2
iii)tan 2
Solution:
As
By pythagoras theorem
So, now we know and (minus sign because theta is in the second quadrant and cos & tan are negative in the second quadrant)
i)
Ans:
ii)
Ans:
iii)
Ans:
Function
Suppose a function , where a and b are constants, it can be easily converted into a single trigonometric function of the form where R is a positive constant and is an acute angle.
Example #2
If we have a function:
We equate it with its trigonometric form:
Expand the right hand side:
Compare the coefficients of and :
To find R we take the magnitude of 7 and 24
To find we do:
°
Ans: Hence the trigonometric form is °