Proving sin4x + 2sin2x = 8sinxcos^3x Using Trigonometric Identities

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In summary, to solve problems involving trigonometric expressions, it is important to remember and practice various formulas such as double and triple angle formulas. Being comfortable with converting between different forms of the same expression is also crucial. With practice, one can easily recognize which form to use in a given problem.
  • #1
Attis
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Homework Statement


show that sin4x + 2sin2x = 8sinxcos^3x



Homework Equations


sin2x =2sinxcosx


The Attempt at a Solution


I started out by letting sin4x = sin(2x*2) so that I could plugg in sin2x = 2sinxcosx in the equation.
sin(2x*2) +2sin2x =
sin2*sin2x +2sin2x =
sin2 * (2sinxcosx) + 2*sin2x =
sin2* (2sinxcosx) + 2* (2sinxcosx)
and I end up with 8sinxcosx
which is wrong!
??
 
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  • #2
Attis said:
sin(2x*2) +2sin2x =
sin2*sin2x +2sin2x =
sin2 * (2sinxcosx) + 2*sin2x =
sin2* (2sinxcosx) + 2* (2sinxcosx)
and I end up with 8sinxcosx
which is wrong!
??

sin(ab) ≠ (sina)(sinb) i.e sin4x≠(sin2)(sin2x)

sin4x = 2sin2xcos2x
 
  • #3
Tanya Sharma said:
sin(ab) ≠ (sina)(sinb) i.e sin4x≠(sin2)(sin2x)

sin4x = 2sin2xcos2x

Ok, did you derive that from sin2x = 2sinxcosx?
In that case, I don´t understand why sin4x = 2sin2xcos2x and not 4sin2xcos2x (i.e twice as much as sin2x = 2sinxcosx?)
 
  • #4
Attis said:
Ok, did you derive that from sin2x = 2sinxcosx?

Yes

Attis said:
In that case, I don´t understand why sin4x = 2sin2xcos2x and not 4sin2xcos2x (i.e twice as much as sin2x = 2sinxcosx?)

Please answer this - What is sin2A ?
 
Last edited:
  • #5
Tanya Sharma said:
Yes



Please answer this - What is sin2A ?

2sinAcosA?

Or is this some sort of a trick question?
 
  • #6
Attis said:
Or is this some sort of a trick question?

Not at all . Just trying to make you understand the formula :smile:

Attis said:
2sinAcosA?

sin2A = 2sinAcosA . Now replace A with 2x on both the sides .What do you get ?
 
  • #7
Tanya Sharma said:
Not at all . Just trying to make you understand the formula :smile:



sin2A = 2sinAcosA . Now replace A with 2x on both the sides .What do you get ?

Ah! I see. Thanks!
sin2*2x= 2sin2xcos2x.
I´ll carry on now and see if I get anywhere.
 
  • #8
Somehow it keeps on getting more and more complicated. Do you have any idea on how I could keep it "simple"?

sin4x + 2sin2x = 8sinxcos^3x
on the left hand side:
2sin2xcos2x + 2sin2x =
2*2sinxcosx(cos^2x - sin^2x) + 2*2sinxcosx =
4sinxcos^3x - 2sin^3xcosx + 4sinxcosx

It really doesn´t feel right...
 
  • #9
Attis said:
2sin2xcos2x + 2sin2x

Is there something common in the two terms ? If yes ,take out the common factor .What do you get ?
 
  • #10
Tanya Sharma said:
Is there something common in the two terms ? If yes ,take out the common factor .What do you get ?

2sin2x(cos2x + 1)?
 
  • #11
Good...

1+cos2x = ?
 
  • #12
1 + cos^2x - sin^2x
?
 
  • #13
Attis said:
1 + cos^2x - sin^2x
?

Correct...Is there some other way of expressing this expression ? Can a couple of terms be combined ?
 
  • #14
There are two options:
1) 1 + cos^2x -sin^2x =
1+cos^2x - (1-cos^2x)=2cos^2x

2) 1+(1-sin^2x)-sin^2x=
1+1 - sin^2x -sin^2x = 2-2sin^2x = 2(1-sin^2x)

?
 
  • #15
Attis said:
There are two options:
1) 1 + cos^2x -sin^2x =
1+cos^2x - (1-cos^2x)=2cos^2x

Right...

Now put sin2x = 2sinxcosx and (1+cos2x) = 2cos2x in the expression you have in post#10.
 
Last edited:
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  • #16
Tanya Sharma said:
Right...

Now put sin2x = 2sinxcosx and (1+cos2x) = 2cos2x in the expression you have in post#10.

Perfect! Now I got it.
Thanks a lot! you´ve been a massive help.
Do you have any general tips on how to solve such problems?
 
  • #17
Remembering trigonometric formulas and practicing variety of problems of increasing complexity is the key .If you remember the formulas then it will help you in manipulating the trigonometric expressions .

You should be comfortable with double ,triple angle formulas .How to convert from one form to other .

Cos2x = cos2x-sin2x = 1-2sin2x = 2cos2x-1 .

sinx = √[(1-cos2x)/2] ; cosx = √[(1+cos2x)/2]

All the above are various ways of writing the same thing .With practice you will recognize which form to use .
 
  • #18
Tanya Sharma said:
Remembering trigonometric formulas and practicing variety of problems of increasing complexity is the key .If you remember the formulas then it will help you in manipulating the trigonometric expressions .

You should be comfortable with double ,triple angle formulas .How to convert from one form to other .

Cos2x = cos2x-sin2x = 1-2sin2x = 2cos2x-1 .

sinx = √[(1-cos2x)/2] ; cosx = √[(1+cos2x)/2]

All the above are various ways of writing the same thing .With practice you will recognize which form to use .

Ok. I´ll do my best. Thanks once again!
 

Related to Proving sin4x + 2sin2x = 8sinxcos^3x Using Trigonometric Identities

What is the equation "sin4x + 2sin2x = 8sinxcos^3x" and why is it important?

The equation "sin4x + 2sin2x = 8sinxcos^3x" is a trigonometric identity that expresses the relationship between the sine, cosine, and tangent functions. It is important because it allows us to simplify and solve complex trigonometric equations and problems.

What are the steps for proving "sin4x + 2sin2x = 8sinxcos^3x" using trigonometric identities?

The steps for proving this identity are as follows:

  1. Start with the left side of the equation and use the double angle identity for sine to rewrite sin4x as 2sin2xcos2x.
  2. Next, use the double angle identity for cosine to rewrite cos2x as 1 - 2sin^2x.
  3. Substitute these values into the equation to get 2sin2xcos2x + 4sin2x = 8sinxcos^3x.
  4. Factor out 2sin2x to get 2sin2x(cos2x + 2) = 8sinxcos^3x.
  5. Use the Pythagorean identity for sine and cosine to simplify the equation to 2sin2x(1 - sin^2x + 2) = 8sinxcos^3x.
  6. Simplify further to get 4sin2x(cos^2x) = 8sinxcos^3x.
  7. Finally, use the double angle identity for cosine to rewrite cos^2x as 1 - sin^2x and you will have proven the identity.

Can this equation be proven using other trigonometric identities?

Yes, there are multiple ways to prove this identity using different trigonometric identities. Some other possible methods include using the half angle identities, the sum and difference identities, and the product-to-sum identities.

What are some real-world applications of this trigonometric identity?

This identity can be used in various fields such as engineering, physics, and astronomy to solve problems involving angles and trigonometric functions. It is also used in computer graphics and animation to create realistic movements and angles in 3D models.

How can I practice and improve my understanding of this identity?

To practice and improve your understanding of this identity, you can solve various trigonometric equations and problems that involve it. You can also create your own practice problems and use online resources such as interactive quizzes and tutorials to test your knowledge and skills.

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