Homework Help for Calculus

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Calculus deals with computation of Mathematical operations. Differential and integral calculus are the most common areas, they study infinitesimal differences and functions with derivatives, respectively.

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in Calculus·
6 Oct 2023

Let's start by solving for p and q using the given equations:

 

Given:

x = va(sinu + cosv)

y = va(cosu - sinv)

z = 1 + sin(u - v)

 

To find p and q, we need to eliminate u and v from the equations. Here's how we can do it:

 

1. Square both sides of the first equation:

x^2 = v^2a^2(sinu + cosv)^2

 

2. Square both sides of the second equation:

y^2 = v^2a^2(cosu - sinv)^2

 

3. Add the squared equations together:

x^2 + y^2 = v^2a^2(sinu + cosv)^2 + v^2a^2(cosu - sinv)^2

 

4. Expand and simplify the equation:

x^2 + y^2 = v^2a^2(sin^2u + 2sinucosv + cos^2v) + v^2a^2(cos^2u - 2sinvcosu + sin^2v)

 

5. Combine like terms:

x^2 + y^2 = v^2a^2(sin^2u + cos^2u + sin^2v + cos^2v) + 2v^2a^2(sinucosv - sinvcosu)

 

6. Simplify further using trigonometric identities:

x^2 + y^2 = v^2a^2 + 2v^2a^2(sin(u + v))

 

7. Now, let's look at the equation for z:

z = 1 + sin(u - v)

 

8. Square both sides of the equation:

z^2 = (1 + sin(u - v))^2

 

9. Expand and simplify the equation:

z^2 = 1 + 2sin(u - v) + sin^2(u - v)

 

10. Substitute the value of z from the original equation:

z^2 = 1 + 2sin(u - v) + sin^2(u - v)

 

11. Simplify further:

z^2 = 1 + 2sin(u - v) + (1 - cos^2(u - v))

 

12. Simplify even more:

z^2 = 2 - cos^2(u - v) + 2sin(u - v)

 

13. Rearrange the equation:

cos^2(u - v) = 2 - z^2 - 2sin(u - v)

 

14. Substitute the value of sin(u - v) from the equation derived in step 6:

cos^2(u - v) = 2 - z^2 - 2v^2a^2

 

15. Take the square root of both sides:

cos(u - v) = ±√(2 - z^2 - 2v^2a^2)

 

16. Now, let's find sin(u - v) using the equation derived in step 6:

sin(u - v) = (x^2 + y^2 - v^2a^2) / (2va^2)

 

17. Substitute the values of cos(u - v) and sin(u - v) into the equation for x:

x = va(sinu + cosv)

 

18. Substitute the values of sin(u - v) and cos(u - v) into the equation for y:

y = va(cosu - sinv)

 

19. Simplify the equations further and solve for p and q:

p = arcsin((x - y) / (2va))

q = arccos((x + y) / (2va))

 

These are the values of p and q based on the given equations. Please note that there may be other solutions or constraints depending on the specific values of x, y, z, v, and a.

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