if given a certain value how would I find what angle does the unit tangent make with the x-axis?
that would be dy/dx
so my unit tangent vector T(w) = cos(w)i+sin(w)j and say w=-pi/3 how would i find what angle does the unit tangent vector T(w) make with the x-axis?
since you have parametric equations,
y = sinw
x = cosw
dy/dx = (dy/dw) / (dx/dw) = cosw/-sinw = -cotw
or note that the equation could have been written
x^2 + y^2 = 1
x + y y' = 0
y' = -x/y = -cosw/sinw = -cotw
and then i would plug in my value for w to find the angle?
oops. sorry. I didn't notice that T was the tangent vector. I took it to be the original position vector. So you must have had
r(w) = sinw i - cosw j
so things are backwards. But you can see how to do it.
my r(w) = wi-ln(cosw)j so r'(w) = i + tanwj. so to find the angle it would be tan(w)
To find the angle that the unit tangent makes with the x-axis, you can follow these steps:
1. Determine the slope of the tangent line at the given point.
2. Use the arctangent function (often denoted as atan or tan^(-1)) to find the angle. This function takes the slope value as input and returns the corresponding angle in radians or degrees.
Here's an example to illustrate the process:
Let's say you have a function f(x) = x^2 + 1 and you need to find the angle that the unit tangent makes with the x-axis when x = 2.
Step 1: Calculate the slope of the tangent line.
To find the slope, you can take the derivative of the function f(x). In this case, the derivative of f(x) = x^2 + 1 is f'(x) = 2x. Evaluate the derivative at x = 2 to get the slope: f'(2) = 2 * 2 = 4.
Step 2: Find the angle using the arctangent function.
Apply the arctangent function to the slope value: angle = atan(4). This will give you the angle in radians. If you want the angle in degrees, you can convert it by multiplying by 180/π (since there are π radians in 180 degrees).
So, to summarize, given a certain value, you can find the angle that the unit tangent makes with the x-axis by calculating the slope of the tangent line and using the arctangent function.