Tangent line approximation with sinx/x + siny/y = C

[franklingroye]

Homework Statement

Suppose the point (pi/3, pi/4) is on the curve sinx/x + siny/y = C, where C is a constant. Use the tangent line approximation to find the y-coordinate of the point on the curve with x-coordinate pi/3 + pi/180.

Homework Equations

TLA: f(a) + f'(a)(x-a)

Where a is the known value of x and x is the value you are estimating the output of.

The Attempt at a Solution

Before differentiating I figured that I could replace siny/y with siny * y^-1.
I got here:

((x)(cosx) – sinx)/x² + y’(siny)(-y^-2) + cosy(y’)(y^-1) = 0

which I THINK can be simplified to:

((x)(cosx) – sinx)/x² + y’(siny*-y^-2 + cosy*y^-1)

dunno where to go with it from there D:

MAYBE, here's what I attempted:

((x)(cosx) - sinx)/x² = -y’(siny*-y^-2 + cosy*y^-1)
-((((x)(cosx) - sinx)/x²)/(siny*-y^-2 + cosy*y^-1)) = y'

But that doesn't seem right. The question gave the curve sinx/x + siny/y = C with the point (pi/3, pi/4) given. Substituting I got C = 1.7273097. However using tangent line approximation with the derivative I calculated I got the y coordinate there to be 418.879:

f(a) + f’(a)(x – a)
f(π/3) + f’(π/3)(π/180)
1.7273097 + 24000(π/180)

I know I could just do normal algebra for this instead of calculus but the instructions say to use tangent line approximation so I think I'd get points docked. I am pretty sure though the answer I'm after is y = 0.905801 based on just solving for the values using C = 1.7273097 and x = pi/3 + pi/180... pretty far off from 418.879.

Someone please help, I've got no idea what to do here.

If it helps get the help any quicker I don't really care about a pretty, formatted answer... I really need help on this soon so whatever helps.

 

[SammyS]

Homework Statement

Suppose the point (pi/3, pi/4) is on the curve sinx/x + siny/y = C, where C is a constant. Use the tangent line approximation to find the y-coordinate of the point on the curve with x-coordinate pi/3 + pi/180.

Homework Equations

TLA: f(a) + f'(a)(x-a)

Where a is the known value of x and x is the value you are estimating the output of.

The Attempt at a Solution

Before differentiating I figured that I could replace siny/y with siny * y^-1.
I got here:

((x)(cosx) – sinx)/x² + y’(siny)(-y^-2) + cosy(y’)(y^-1) = 0

which I THINK can be simplified to:

((x)(cosx) – sinx)/x² + y’(siny*-y^-2 + cosy*y^-1)

dunno where to go with it from there D:

MAYBE, here's what I attempted:

((x)(cosx) - sinx)/x² = -y’(siny*-y^-2 + cosy*y^-1)
-((((x)(cosx) - sinx)/x²)/(siny*-y^-2 + cosy*y^-1)) = y'

But that doesn't seem right. The question gave the curve sinx/x + siny/y = C with the point (pi/3, pi/4) given. Substituting I got C = 1.7273097. However using tangent line approximation with the derivative I calculated I got the y coordinate there to be 418.879:

f(a) + f’(a)(x – a)
f(π/3) + f’(π/3)(π/180)
1.7273097 + 24000(π/180)
...

Hello franklingroye. Welcome to PF.

I agree with your result for C.

What is your result for y' when x = π/3 and y = π/4 ?

I got y' ≈ -0.000876215 there.

The way you are using the tangent line approximation is wrong.

y = f(x), so f(π/3) = π/4

 

[franklingroye]

Hello franklingroye. Welcome to PF.

I agree with your result for C.

What is your result for y' when x = π/3 and y = π/4 ?

I got y' ≈ -0.000876215 there.

The way you are using the tangent line approximation is wrong.

y = f(x), so f(π/3) = π/4

Goodness, I can't believe I managed to mess up with f(x) haha.

I'm not sure how to really get y' out of this is the issue I am having. If I leave it as-is instead of changing it to siny*y^-1:

((x)(cosx) – sinx)/x² + y'((y)(cosx) – siny)/y²
((x)(cosx) – sinx)/x² = -y'((y)(cosy) – siny)/y²
[((x)(cosx) – sinx)/x²]/(y)(cosy) – siny)/y² = -y'
-[((x)(cosx) – sinx)/x²]/(y)(cosy) – siny)/y² = y'
-((((pi/3)(cos(pi/3)) - sin(pi/3))/(pi/3)^2)/((((pi/4))(cos(pi/4)) - sin(pi/4))/(pi/4)^2)) = -1.269321

I've got no idea where I'm screwing up

I tested the TLA equation with y' = -0.000876215 and it's correct but now I just need to figure out how to get there. :\ I'm obviously messing up somewhere in the differentiation process.

 

[SammyS]

I checked your work to here and it looked good.

((x)(cosx) - sinx)/x² = -y’(siny*(-y^-2) + cosy*y^-1)​

Solving that for y', I got:

\displaystyle y'=-\frac{y^2\left(x\cos(x)-\sin(x)\right)}{x^2\left(y\cos(y)-\sin(y)\right)}​

And I see that I had a sign error when I calculated y' at the given point.

It should have been y' ≈ 0.00728974

(Yes, it's positive.)

 

[franklingroye]

I checked your work to here and it looked good.

((x)(cosx) - sinx)/x² = -y’(siny*(-y^-2) + cosy*y^-1)​

Solving that for y', I got:

\displaystyle y'=-\frac{y^2\left(x\cos(x)-\sin(x)\right)}{x^2\left(y\cos(y)-\sin(y)\right)}​

And I see that I had a sign error when I calculated y' at the given point.

It should have been y' ≈ 0.00728974

(Yes, it's positive.)

I don't get what's going on here, using my calculator to simply plug in pi/3 and pi/4 for x/y in your derivative I end up with 0.15257. ,-,

 

[SammyS]

I don't get what's going on here, using my calculator to simply plug in pi/3 and pi/4 for x/y in your derivative I end up with 0.15257. ,-,

Check parentheses.

After rechecking my algebra, I now get y' ≈ -1.26932145 ; back to a negative, and it should be negative when considering the values of x & y used in

\displaystyle y'=-\frac{y^2\left(x\cos(x)-\sin(x)\right)}{x^2\left(y\cos(y)-\sin(y)\right)}​

For the values of x & y used here, cos(x)=1/2, sin(x)=(√3)/2, cos(y)=sin(y)=(√2)/2 .