what is the linear approximation of sin(0.11)?
I am confused because there is not a point on the unit circle that is close to that value? Where would I start?
in radians sin theta = theta for small angles
so if 0.11 is in radians then the answer is 0.11
however if it is 0.11 degrees, then you must convert
0.11 degrees * 3.14159 radians / 180 degrees = 0.001919 radians
so if degrees then sin 0.11 degrees = about 0.002
you want the tangent line at x = 0.11
since y' = cosx, that means you want the line
y - sin(0.11) = cos(0.11)(x - 0.11)
or, approximately
y - 0.11 = 0.994(x - 0.11)
To find the linear approximation of sin(0.11), you can use the concept of linear approximation or tangent line approximation.
The first step is to find a point on the unit circle that is close to the given value, in this case, 0.11. Since 0.11 is not directly on the unit circle, you can look for values that are close to it.
One approach is to convert 0.11 to degrees. In radians, there are approximately 2π radians in a full circle, which is equivalent to 360 degrees. Hence, you can convert 0.11 to degrees by multiplying it by 360:
0.11 * 360 ≈ 39.6 degrees
Now you can look for the angle on the unit circle that is closest to 39.6 degrees. In this case, the closest angle is 40 degrees or π/9 radians.
Next, you can calculate the exact value of sin(π/9) using trigonometric functions. In this case, sin(π/9) is approximately 0.3473.
Now that we have a point on the unit circle (π/9, 0.3473) that is close to the given input of sin(0.11), we can approximate sin(0.11) using linear approximation.
To do this, you can consider the tangent line to the unit circle at the point (π/9, 0.3473). The equation of the tangent line can be found using the point-slope form:
(y - y1) = m(x - x1)
In this case, (x1, y1) is the point (π/9, 0.3473) and m is the slope of the tangent line. The slope of the tangent line is equal to the derivative of sin(x) at x = π/9, which can be found using calculus:
d/dx(sin(x)) = cos(x)
cos(π/9) ≈ 0.9397
Now we can plug in the values into the equation:
(y - 0.3473) = 0.9397(x - π/9)
Finally, we can substitute the value x = 0.11 into the equation to find the linear approximation:
(y - 0.3473) = 0.9397(0.11 - π/9)
Simplifying this equation will give you the linear approximation of sin(0.11).