These questions are related to de moivre's theorem:
z^n + 1/z^n = 2cosntheta
z^n - 1/z^n = 2 isin ntheta
1. Express sin^5theta in the form Asintheta + Bsin3theta + Csin5theta and hence find the integral of sin^5theta.
2. Express sin^6theta in multiples of costheta and hence evaluate the integral of sin^6theta between 0 and pi/2.
3. Evaluate the integral of cos^7theta between pi/2 and 0.
The answers are:
1. -1/80cos5theta + 5/48cos3theta - 5/8costheta
2. 5pi/32
3. 5pi/32
Could you please guide me through the solutions, thank you SO much :)
If you had posted one at I time, I could have helped with one of them. It is past my bedtime here.
When you say "between pi/2 and 0" do you mean from pi/2 to zero? That would be backwards.
Please be more specific.
Sorry, yes I meant from 0 to pi/2 for the last two questions.
Certainly! Let's go through each question step by step:
1. Express sin^5theta in the form Asintheta + Bsin3theta + Csin5theta and hence find the integral of sin^5theta.
To express sin^5theta in the desired form, we'll use the trigonometric identity: sin(2theta) = 2sin(theta)cos(theta).
We can rewrite sin^5theta as (sin^2theta)^2 * sintheta. Then,
(sin^2theta)^2 * sintheta = (1 - cos^2theta)^2 * sintheta.
Expanding (1 - cos^2theta)^2 gives us:
(1 - 2cos^2theta + cos^4theta) * sintheta = sintheta - 2cos^2theta * sintheta + cos^4theta * sintheta.
Now, let's focus on the terms one by one:
- sintheta can be expressed as sintheta = sintheta.
- To express -2cos^2theta * sintheta in terms of sintheta, we'll use the identity: 2cos^2theta = 1 + cos(2theta). Therefore, -2cos^2theta * sintheta becomes -2(1 + cos(2theta)) * sintheta.
- To express cos^4theta * sintheta in terms of sintheta, we'll use the identity: cos^4theta = (1/8)(3 + 4cos(2theta) + cos(4theta)). Substituting this into cos^4theta * sintheta gives us (1/8)(3 + 4cos(2theta) + cos(4theta)) * sintheta.
Combining all the terms, we have:
sin^5theta = sintheta - 2(1 + cos(2theta)) * sintheta + (1/8)(3 + 4cos(2theta) + cos(4theta)) * sintheta.
To find the integral of sin^5theta, integrate each term separately. The integral of sintheta is -cos(theta), the integral of -2(1 + cos(2theta)) * sintheta is 2sintheta + (1/2)sin(2theta), and the integral of (1/8)(3 + 4cos(2theta) + cos(4theta)) * sintheta can be found by integrating term by term.
Finally, simplify the expression to obtain the answer.
2. Express sin^6theta in multiples of costheta and hence evaluate the integral of sin^6theta between 0 and pi/2.
To express sin^6theta in terms of costheta, we'll use the identity: sin^2theta = 1 - cos^2theta.
Start by rewriting sin^6theta as (sin^2theta)^3. Substituting 1 - cos^2theta for sin^2theta, we have:
(sin^2theta)^3 = (1 - cos^2theta)^3.
Expand (1 - cos^2theta)^3, and we get:
1 - 3cos^2theta + 3cos^4theta - cos^6theta.
Now, let's focus on each term:
- 1 can be expressed as 1costheta.
- To express -3cos^2theta in terms of costheta, we'll use the identity: cos^2theta = 1 - sin^2theta. Therefore, -3cos^2theta becomes -3(1 - sin^2theta) = -3 + 3sin^2theta, which can be further simplified to 3sin^2theta - 3.
- To express 3cos^4theta in terms of costheta, we'll use the identity: cos^4theta = (1/8)(3 + 4cos(2theta) + cos(4theta)). Substituting this into 3cos^4theta gives us 3 * (1/8)(3 + 4cos(2theta) + cos(4theta)).
- cos^6theta can be expressed as (1/64)(35 + 30cos(2theta) + 15cos(4theta) + cos(6theta)).
Combining all the terms, we have:
sin^6theta = 1costheta + (3sin^2theta - 3) + 3 * (1/8)(3 + 4cos(2theta) + cos(4theta)) + (1/64)(35 + 30cos(2theta) + 15cos(4theta) + cos(6theta)).
To find the integral of sin^6theta between 0 and pi/2, integrate each term separately. The integral of 1 with respect to theta is theta. The integral of 3sin^2theta - 3 can be found by using the identity: sin^2theta = (1/2)(1 - cos(2theta)), and integrating term by term for the remaining terms.
Evaluate the integral from 0 to pi/2, and simplify the expression to obtain the answer.
3. Evaluate the integral of cos^7theta between pi/2 and 0.
To evaluate the integral of cos^7theta between pi/2 and 0, we'll use the substitution method. Let u = sintheta, then du = costheta dtheta.
The integral becomes:
∫(cos^7theta * costheta) dtheta = ∫(cos^8theta) du.
Now, we integrate only the u component since it is in terms of u, not theta:
∫(u^8) du = (1/9)u^9.
Substituting u back in terms of sintheta, the integral becomes:
(1/9)sin^9theta.
Evaluate the integral from pi/2 to 0 by plugging in the values, and simplify the expression to obtain the answer.
I hope this helps! Let me know if you have any further questions.