Question

It is a common approximation in physics to say that sin x ≈ x near 0. Use a tangent line
approximation at 0 to show that this is a good approximation. How much error is there in this
approximation at x = π/6? Please help I do not understand this at all

Answer 1

So it sounds like we are to find the tangent line to y=sin(x) at x=0

Answer 2

Yes and then at pi/6 I think

Answer 3

Are we suppose to find the tangnet line at pi/6?
I don't see that?
I see we are to use the approximation we found from the tangent line to approximate sin(pi/6)

Answer 4

but maybe I'm wrong.
i just don't see where it says to find the tangent line at pi/6

Answer 5

anyways
which part do you need help on

Answer 6

At the end it does but it turned into question marks sorry. I need help with all of it

Answer 7

so you need help finding the tangent line to y=sin(x) at x=0?

Answer 8

well first of all a line looks like
y-y1=m(x-x_1)
assuming it isn't vertical anyways

Answer 9

m=f'(0)
(x1,y1)=(0,sin(0))

Answer 10

find f' first

Answer 11

So is f'(0) is sin x=x

Answer 12

remember we are trying to find the tangent line to y=sin(x) at x=0
we haven't establish the approximation sin(x)=x yet

Answer 13

so we can't use that yet

Answer 14

y=sin(x)
so
y'=?

Answer 15

do you know the derivative of sin(x)?

Answer 16

yes its cos x

Answer 17

so y=sin(x) means y'=cos(x)
now we just need evaluate y' at x=0 to find the slope at x=0

Answer 18

\[y-y_1=f'(0)(x-x_1)\]
\[y-y_1=\cos(0)(x-x_1) \\ \text{ and since again we are finding the tangnet line \to f(x)=\sin(x) } \\ \text{ at (0,\sin(0)) then we have the tangnet line is } \\ y-\sin(0)=\cos(0)(x-0)\]
I will let you simplify that
when you have simplify things
you will see how they got the approximation sin(x)=x

Answer 19

Okay so how do I get to the pi/6

Answer 20

you have approximation is sin(x)=x
and sin(pi/6) is the exact value
pi/6 is the approximation to sin(pi/6)
using the sin(x)=x approximation for values near x=0

Answer 21

\[relative error=\frac{x_*-x}{x} \]
where x* is the approximation to the actual value x

Answer 22

you can also find the absolute error
\[x_*-x\]

Answer 23

they are probably just looking for absolute error

Answer 24

Okay so what do I plug in for that equation? I'm sorry if I sound really dumb

Answer 25

sin(pi/6) is the actual value
pi/6 is the approximation...

Answer 26

I called x the actual and x_* the approximation

Answer 27

so I just plug in those numbers
pi/6-sin(pi/6)=absolute error

Answer 28

Okay makes more sense. Thank you :)

Answer 29

now I don't see where in the question it says find the tangent line to y=sin(x) at pi/6
you can if you want to but I think you would be doing extra work

to me i read this as
find the the tangnet line to f(x)=sin(x) at x=0
and use that approximation sin(x)=x for x=pi/6 just to test how good the approximation is for values near x=0