(cosx)^2 half angle formula

In summary, we are given the equation y=sin2x bounded by x=0,x=pi,y=0 and asked to find the volume when this region is revolved around the x-axis. Using the cross section A=pi(sin2x)^2 and the substitution u=sinx with du=cosxdx, we can use the half-angle formula cos(2x)=1-2sin2x to simplify the integral. Applying the Pythagorean identity and the half-angle formula again, we can easily integrate the resulting expression.
  • #1
nameVoid
241
0
y=sin2x bounded by x=0,x=pi,y=0 , revolved around the x-axis

cross section A=pi(sin2x)^2

latex2png.2.php?z=100&eq=pi%5Cint_%7B0%7D%5E%7Bpi%2F2%7D%20(sin(2x))%5E2.jpg


latex2png.2.php?z=100&eq=4pi%5Cint_%7B0%7D%5E%7Bpi%2F2%7Dsin%5E2xcos%5E2x.jpg

taking u = sinx ; du=cosxdx

im unclear on how to proceed in this case where du needs to satisfy (cosx)^2
the problem hints to use a half angle formula
 
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  • #2


Try the identity cos(2x) = 1 - 2sin2(x). It can be rearranged to be a half-angle formula. :smile:
 
  • #3


still i am unclear on how to proceed here
 
  • #4


nameVoid said:
still i am unclear on how to proceed here

Try using the Pythagorean identity to get sin2(x) - sin4(x), then apply the half angle formula to get cosines (twice for the second term) that are not squared.
 
  • #5


slider142 is trying to say that since sin^2(x)=(1-cos(2x))/2, sin^(2x)=(1-cos(4x))/2. That's pretty easy to integrate.
 
  • #6


perfect
 

Related to (cosx)^2 half angle formula

1. What is the (cosx)^2 half angle formula?

The (cosx)^2 half angle formula is a trigonometric identity that expresses the square of the cosine of an angle in terms of the angle's half angle. It is written as cos^2(x/2) = (1 + cosx)/2.

2. How is the (cosx)^2 half angle formula derived?

The (cosx)^2 half angle formula can be derived using the double angle formula for cosine, which states that cos2x = 2cos^2x - 1. By substituting x/2 for x in this formula, we can rearrange it to get cos^2(x/2) = (1 + cosx)/2.

3. What is the importance of the (cosx)^2 half angle formula?

The (cosx)^2 half angle formula is important because it allows us to simplify and solve complex trigonometric equations and identities. It is also used in various applications, such as in calculating the area of a triangle or the volume of a cone.

4. How is the (cosx)^2 half angle formula used in real life?

The (cosx)^2 half angle formula has many real life applications, particularly in fields such as engineering, physics, and astronomy. For example, it is used in calculating the trajectory of an object launched at an angle, in determining the strength of a force acting at an angle, and in measuring the position of stars and planets.

5. Are there any other trigonometric identities related to the (cosx)^2 half angle formula?

Yes, there are several other trigonometric identities that are related to the (cosx)^2 half angle formula. These include the (sinx)^2 half angle formula, the (tanx)^2 half angle formula, and the (secx)^2 half angle formula. These identities can be derived from the (cosx)^2 half angle formula using basic trigonometric relationships.

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