iii) Consider
=
But
=
=
=
Alternative: Using
=
Dividing numerator and denominator by cos3θ
Applications of the double and triple formulae
A. Proving Identities
Examples: Prove the following identities
(i)
+
(ii)
=
(iii)
Solution (i)
I. Proof
Dealing with L.H.S
=
= cos2A + cos2A – sin2A
= 2cos2A – sin2A
= 2cos2A – sin2A
–
= 2 –
=
R.H.S
=
II. Solution (ii)
Dealing with L.H.S
But
A =
R.H.S
III. Solution (iii)
=
=
=
=
Work on the following problems to prove the identities:
- i)
=
- ii)
=
- iii)
- iv)
=
- v)
+
= 2
- vi)
=
- vii)
=
- viii)
=
- ix)
= 2
- x)
=
- xi)
+
=
Warm up with:
i) Find tan
without calculating mathematical tables
ii)
HALF ANGLES FORMULAE
From
=
–
Then
=
=
=
=
1 –
=
=
– 1 +
=
=
=
Again from
=
–
But
=
1 –
=
1 –
1 –
–
1 +
=
2
2
=
1 –
For
=
=
=
=
Similarly, the formulae can be expressed as
EQUATION OF THE FORM
a
=
c
where a, b, and c are real constants.
The task here is to solve the equation. There are two ways to solve:
i. Using t-formulae
ii. Using R-formula (or transforming a function a
+ b
= c as a single function)
7 Comments