a) Find a linear approximation of y=sinx at x=pi/6
b) use part (a) to approximate sin(61pi/360) and sin(59pi/360)
I just really have no idea how to approach this problem. I know the formula is y=f(a)+f'(a)(x-a). Does that mean it would be (sin(pi/6))+(0)((pi/6)-(sin(pi/6))? Any help would be appreciated.
"a) Find a linear approximation of y=sinx at x=pi/6 "
no linear approximation needed here, you MUST know that sin(π/6) = 1/2
for b) sin(61π/360)
= sin(60π/360 + π/360)
= sin(π/6 + (π/360)
so we will now use the formula that you stated:
f(x+a)=f(a)+f'(a)(x-a)
recall that if f(x) = sinx, then f '(x) = cosx
so let x = π/6 and a = π/360
sin(61π/360) = sin(π/6) + cos(π/6)*(π/360)
= 1/2 + (√3/2)(π/360)
= .507557
real answer: sin(61π/360) = .507538
error of .000019 , not bad
for the second part, note that 59π/360
= π/6 - π/360
so the last factor would be -π/360
and you should get .49244..
compared to the real answer of .49242..
(again, very close)
a) To find the linear approximation of y = sin(x) at x = π/6, we need to compute the value of y and its derivative at x = π/6.
We have y = sin(x) and we need to find y(π/6), which is sin(π/6):
y(π/6) = sin(π/6) = 1/2
Now, we need to find the derivative of y = sin(x) and evaluate it at x = π/6.
y'(x) = cos(x)
y'(π/6) = cos(π/6) = √3/2
Using the formula for linear approximation,
y ≈ f(a) + f'(a)(x - a)
We have a = π/6, y = sin(x), f(a) = sin(π/6) = 1/2, f'(a) = cos(π/6) = √3/2, and x - a = x - π/6.
Therefore, the linear approximation of y = sin(x) at x = π/6 is:
y ≈ 1/2 + (√3/2)(x - π/6)
b) Now, we can use the linear approximation found in part (a) to approximate sin(61π/360) and sin(59π/360).
For sin(61π/360), we substitute x = 61π/360 into the linear approximation:
y ≈ 1/2 + (√3/2)(61π/360 - π/6)
Simplifying,
y ≈ 1/2 + (√3/2)(61π - 60π)/360
y ≈ 1/2 + (√3/2)(π/360)
Using a calculator to evaluate (√3/2)(π/360), we find that it is approximately 0.009512.
Therefore, sin(61π/360) is approximately 1/2 + 0.009512.
Similarly, for sin(59π/360), we substitute x = 59π/360 into the linear approximation:
y ≈ 1/2 + (√3/2)(59π/360 - π/6)
y ≈ 1/2 + (√3/2)(59π - 60π)/360
y ≈ 1/2 + (√3/2)(-π/360)
Using a calculator to evaluate (√3/2)(-π/360), we find that it is approximately -0.009512.
Therefore, sin(59π/360) is approximately 1/2 - 0.009512.
To find the linear approximation of y = sin(x) at x = pi/6, you are correct in using the formula y = f(a) + f'(a)(x - a).
a) Let's calculate the linear approximation:
First, substitute a = pi/6 and f(a) = sin(pi/6) into the formula.
So, y = sin(pi/6) + f'(pi/6)(x - pi/6).
Now, we need to find f'(pi/6), which represents the derivative of sin(x) evaluated at x = pi/6. The derivative of sin(x) is cos(x). Therefore, f'(pi/6) = cos(pi/6) = √3/2.
Substituting these values into our formula, we get:
y = sin(pi/6) + (√3/2)(x - pi/6).
Simplifying this gives us the linear approximation: y = 1/2 + (√3/2)(x - pi/6).
b) Now, to approximate sin(61pi/360) and sin(59pi/360) using the linear approximation:
For sin(61pi/360):
Plug x = 61pi/360 into the linear approximation equation:
y = 1/2 + (√3/2)(61pi/360 - pi/6).
For sin(59pi/360):
Plug x = 59pi/360 into the linear approximation equation:
y = 1/2 + (√3/2)(59pi/360 - pi/6).
Simplify the expressions above to get the respective approximations for sin(61pi/360) and sin(59pi/360).