Name:
Period:
Course: Math IV
Lesson: 4
Double and half angle derivations
Unit: 4
Date:
Double Angle Formulas
You will now be deriving the double angle formulas. They are probably the easiest
to prove. Their proofs are based on the addition formulas. We will use the fact that
x + x = 2x for each derivation.
Fill in the blanks in this derivation:
sin 2x = sin (___ + ___) = _______x _______x + ______x ______x = ____ sin x cos x
cos 2x = cos (___ + ___) = _______ x _______ x - _______x _______x = ______________________
tan 2x = tan (___ + ____) =
_________+ _________
1− _________∙__________
=
2 ______________
1−__________________
Cosine Double Angle – Revisited
There are alternate forms of the double angle formula that will be useful to us as we
derive the half-angle formulas. These alternate forms are based on the identity
𝑠𝑖𝑛2 𝑥 + 𝑐𝑜𝑠 2 𝑥 = 1. Fill in the blanks below to form two equivalent equations.
1 – sin2x = __________
and
1 – cos2x = __________
Using these substitutions, we can rewrite our cosine double angle formula.
cos 2x = cos2x – sin2x
= _________ – sin2x
= 1 - ____ sin2x
cos 2x = cos2x – sin2x
= cos2x – __________
= ____cos2x - 1
Half-Angle Formulas
We are going to need to do a bit of algebra to derive our half-angle formulas for sine
and cosine. Fill in the blanks as we rearrange the cosine double angle formulas to
solve for sin x and cos x.
Solve for sin x:
cos 2x = 1 – 2sin2x
Solve for cos x:
cos 2x = 2cos2x - 1
Name:
Period:
Course: Math IV
Lesson: 4
Double and half angle derivations
Unit: 4
2sin2x = 1 - ________
sin2x =
Date:
1 + _______ = 2cos2x
1−cos 2𝑥
1+cos 2𝑥
sin x = _________
= cos2x
________ = cos x
We haven’t mentioned anything about half-angles yet, but x is half of 2x, so we now
have our formula. Let’s change to another variable so this is less confusing. Let’s let
a
2x = a, then x = 2, and we can write our half angle formulas for sine and cosine:
𝑎
1−𝑐𝑜𝑠𝑎
𝑠𝑖𝑛 2 = ±√
2
and
𝑎
1+𝑐𝑜𝑠𝑎
𝑐𝑜𝑠 2 = ±√
2
The + sign is used differently in this context than we are solving equations. You will
choose either the positive or negative square root, not both, depending on the
quadrant of the original angle. You will see examples very soon.
We’ll turn now to our tangent identity to find a half-angle formula for tangent.
𝑡𝑎𝑛
𝑎
1 − 𝑐𝑜𝑠𝑎
= ±√
2
1 + cos 𝑎
You will sometimes see alternate forms of this formula, however, and they are listed
below.
𝑎
sin 𝑎
1 − cos 𝑎
𝑡𝑎𝑛 =
=
2
1 + cos 𝑎
sin 𝑎
We will not derive those formulas now, but you may see a nice proof at
http://oakroadsystems.com/twt/double.htm#TanHalf
That was a great mathematical work out! I hope you find it easier to
remember these formulas now that you know why they work!