Sin^4Ө =3/8-3/8cos(2Ө) Prove the following trigonometric identity

In summary, the trigonometric identity sin^4(\theta)=3/8-3/8cos(2Ө) does not have a definitive answer when applied to the equation sin^2(\theta)=1-cos^2(\theta).
  • #1
bubbly616
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Homework Statement


Prove the following trigonometric identity. The question is sin^4Ө =3/8-3/8cos(2Ө)

Homework Equations


I think I'm supposed to use the power reducing formulas for trigonometric identities which are
sin^2(u)= (1- cos(2u))/2
cos^2(u)=(1+cos(2u))/2
*Let u represent any integer/value*

The Attempt at a Solution


To expand the equation I separated the equation into (1- cos^2Ө)(1- cos^2Ө) = 1- 2cos^2Ө + (cos^2Ө)^2. I reduced it because sin^2Ө = 1 – cos^2Ө and after this I'm confused on what the next steps are.
 
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  • #2
So, a much better post than the first try. Good !

Your other entry is from the righthand side: what have you got for ##\cos(2\theta)## that might be useful here ?

[edit] ah: your second relevant equation !
 
  • #3
So you applied the sin^2(u) to the lefthand side of the equation and got (1/4) * (1 - cos(2u) )^2 right?

Next expand the

(1/4)*(1 - cos(2u) )^2 = (1/4) * ( 1 - 2*cos(2u) - cos(2u)^2 )

next apply the cos^2(u) rule to the last term and see what you get.
 
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  • #4
As well as the power reducing formulas, you'll want to glance at the double angle formulas.
 
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  • #5
bubbly616 said:

Homework Statement


Prove the following trigonometric identity. The question is sin^4Ө =3/8-3/8cos(2Ө)

Homework Equations


I think I'm supposed to use the power reducing formulas for trigonometric identities which are
sin^2(u)= (1- cos(2u))/2
cos^2(u)=(1+cos(2u))/2
*Let u represent any integer/value*

The Attempt at a Solution


To expand the equation I separated the equation into (1- cos^2Ө)(1- cos^2Ө) = 1- 2cos^2Ө + (cos^2Ө)^2. I reduced it because sin^2Ө = 1 – cos^2Ө and after this I'm confused on what the next steps are.
But now I see a problem coming up:
let ##\theta = \pi/2\ \ ## then ##(sin\theta)^4 = 1\ \ ## and ##\ \ 3/8-3/8\cos(2\theta)=3/4## !?
So no identity at all !
Or did I read the original thingy in the wrong way ?
 
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  • #6
BvU said:
But now I see a problem coming up:
let ##\theta = \pi/2\ \ ## then ##(sin\theta)^4 = 1\ \ ## and ##\ \ 3/8-3/8\cos(2\theta)=3/4## !?
So no identity at all !
Or did I read the original thingy in the wrong way ?
I believe the identity ought to read ##\sin^4(\theta)=\frac 38-\frac 12\cos(2\theta)+\frac 18\cos(4\theta)##.
Looks like someone turned that ##4\theta## into another ##2\theta##.
 
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  • #7
Why would u have to be an integer?
 

1. What is the purpose of proving trigonometric identities?

Proving trigonometric identities is important because it allows us to verify the relationships between different trigonometric functions and to use those relationships to simplify complex expressions.

2. How do you prove a trigonometric identity?

To prove a trigonometric identity, we use algebraic manipulations and properties of trigonometric functions to transform one side of the equation into the other.

3. What are the most common trigonometric identities used in proofs?

The most common trigonometric identities used in proofs are the Pythagorean identities, sum and difference identities, double angle identities, and half angle identities.

4. How can we verify the identity Sin^4Ө =3/8-3/8cos(2Ө)?

We can verify the identity by substituting different values for Ө and using a calculator or trigonometric tables to calculate the values of both sides of the equation. If the values are equal for all values of Ө, the identity is proven.

5. Are there any tips for proving trigonometric identities?

Yes, here are a few tips for proving trigonometric identities: 1) Start with the more complex side of the equation and simplify it using algebraic manipulations and identities, 2) Use Pythagorean identities to change trigonometric functions into their squared forms, and 3) Look for patterns or similarities in the expressions on both sides of the equation.

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