^{2}θ-sin

^{2}θ

^{3}θ - 3cosθsin

^{2}θ

^{4}θ - 6cos

^{2}θsin

^{2}θ + sin

^{2}θ

*n*θ.

(HINT: You will need to look at Pascal's Triangle.)

1) given the double angle identity for cosine: cos2θ=cos^{2}θ-sin^{2}θ

Prove: cos3θ = cos^{3}θ - 3cosθsin^{2}θ

and: cos4θ = cos^{4}θ - 6cos^{2}θsin^{2}θ + sin^{2}θ

2) Continue this process, finding an identity for cos5θ and cos6θ.

3) By examining the above identities (and a few more if needed), try to identify a pattern in order to generalize an identity for cos*n*θ.

(HINT: You will need to look at Pascal's Triangle.)

Tutors, sign in to answer this question.

1) cos(3θ)

= cos(2θ+θ)

= cos2θ cosθ - sin2θ sinθ

= (cos^2 θ - sin^2θ)cosθ - 2sin^2θ cosθ

= cos^3θ - 3cosθ sin^2θ

cos4θ

= cos^2(2θ) - sin^2(2θ)

= (cos^2θ-sin^2θ)^2 - 4sin^2θ cos^2θ

= cos^4θ - 6cos^2θ sin^2θ + sin^4θ <==You had a typo here.)

You can follow this way to get it done. But it is lot simpler if you use De Moivre's theorem.

cosnθ + i sinnθ = (cosθ + i sinθ)^n

Expand and compare real parts:

cosnθ

= cos^nθ - nC2cos^(n-2)θsin^2θ + nC4cos^(n-4)θsin^4θ -...+

= ∑{i = 0, n}nCi cos^(n-i) θ sin^i θ cos(0.5(n-i)pi)

prove: cos(3θ) = cos^3(θ) - 3 cos(θ) sin^2(θ)

and: cos(4θ) = cos^4(θ) - 6 cos^2(θ) sin^2(θ) + sin^2(θ)

2) Continue this process, finding an identity for cos(5θ) and cos(6θ).

3) By examining the above identities (and a few more if needed), try to identify a pattern in order to generalize an identity for cos^n(θ).

1) cos(3θ) = cos(2θ + θ) = cos(2θ)cos(θ) - sin(2θ)sin(θ)

= (cos^2(θ) - sin^2(θ)) cos(θ) - (2 sin(θ) cos(θ)) sin(θ)

= cos^3(θ) - sin^2(θ) cos(θ) - 2 sin^2(θ) cos(θ)

= cos^3(θ) - 3 sin^2(θ) cos(θ)

= cos^3(θ) - 3 cos(θ) sin^2(θ)

cos(4θ) = cos(2θ + 2θ) = cos(2θ)cos(2θ) - sin(2θ)sin(2θ)

= (cos^2(θ) - sin^2(θ)) (cos^2(θ) - sin^2(θ))

- (2 sin(θ) cos(θ)) (2 sin(θ) cos(θ))

= cos^4(θ) - 2 cos^2(θ) sin^2(θ) + sin^4(θ) - 4 cos^2(θ) sin^2(θ)

= cos^4(θ) - 6 cos^2(θ) sin^2(θ) + sin^4(θ)

Note that last term is sin^4(θ), NOT sin^2(θ).

GeoGebra graphs confirm that the correct term IS sin^4(θ).

2) cos(5θ) = cos(3θ + 2θ) = cos(3θ)cos(2θ) - sin(3θ)sin(2θ)

sin(3θ) = sin(2θ + θ) = sin(2θ)cos(θ) + cos(2θ)sin(θ)

= (2 sin(θ) cos(θ)) cos(θ) + (cos^2(θ) - sin^2(θ)) sin(θ)

= 3 sin(θ) cos^2(θ) - sin^3(θ)

cos(5θ) = (cos^3(θ) - 3 cos(θ) sin^2(θ)) (cos^2(θ) - sin^2(θ))

- (3 sin(θ) cos^2(θ) - sin^3(θ)) (2 sin(θ) cos(θ))

= cos^5(θ) - cos^3(θ) sin^2(θ) - 3 cos^3(θ) sin^2(θ)

+ 3 cos(θ) sin^4(θ) - 6 sin^2(θ) cos^3(θ) + 2 sin^4(θ) cos(θ)

= cos^5(θ) - 10 cos^3(θ) sin^2(θ) + 5 cos(θ) sin^4(θ)

cos(6θ) = cos(3θ + 3θ) = cos(3θ)cos(3θ) - sin(3θ)sin(3θ)

= (cos^3(θ) - 3 cos(θ) sin^2(θ))^2 - (3 sin(θ) cos^2(θ) - sin^3(θ))^2

= cos^6(θ) - 6 cos^4(θ) sin^2(θ) + 9 cos^2(θ) sin^4(θ)

- 9 sin^2(θ) cos^4(θ) + 6 sin^4(θ) cos^2(θ) - sin^6(θ)

= cos^6(θ) - 15 cos^4(θ) sin^2(θ) + 15 cos^2(θ) sin^4(θ) - sin^6(θ)

3) cos(2θ) = cos^2(θ) - sin^2(θ)

cos(3θ) = cos^3(θ) - 3 cos(θ) sin^2(θ)

cos(4θ) = cos^4(θ) - 6 cos^2(θ) sin^2(θ) + sin^4(θ)

cos(5θ) = cos^5(θ) - 10 cos^3(θ) sin^2(θ) + 5 cos(θ) sin^4(θ)

cos(6θ) = cos^6(θ) - 15 cos^4(θ) sin^2(θ) + 15 cos^2(θ) sin^4(θ) - sin^6(θ)

Pascal’s Triangle:

1 0 0 0 0 0 0

1 1 0 0 0 0 0

1 2 1 0 0 0 0

1 3 3 1 0 0 0

1 4 6 4 1 0 0

1 5 10 10 5 1 0

1 6 15 20 15 6 1

Let P(r,c) be the element in row r and column c of Pascal’s Triangle. The second column contains the row number.

Then cos(n θ) = cos^n(θ) - P(n,3) (cos(θ))^(n-2) sin^2(θ) + P(n,5) (cos(θ))^(n-4) sin^4(θ) - P(n,7) (cos(θ))^(n-6) sin^6(θ) + . . .

Already have an account? Log in

By signing up, I agree to Wyzant’s terms of use and privacy policy.

Or

To present the tutors that are the best fit for you, we’ll need your ZIP code.

Your Facebook email address is associated with a Wyzant tutor account. Please use a different email address to create a new student account.

Good news! It looks like you already have an account registered with the email address **you provided**.

It looks like this is your first time here. Welcome!

To present the tutors that are the best fit for you, we’ll need your ZIP code.

Please try again, our system had a problem processing your request.