Given the curve descibed by the equation r = 3 cos theta, find the angle that the tangent line makes with?

Question

the radus vector when theta = 120°

Answer 1

If we want to know what the angle is between a line from the origin to pt. (x,y) on a curve f(x), and the tangent to the curve at that same point, then:
The slope of the line from the origin = y/x = tan(A)
The slope of the tangent line is f'(x) = tan(B)

The angle between them is:
B - A
(To see that, you could draw a graph of a parabola).

Remember x=r cos θ and y = r sin θ
Since θ = 120°
and r = 3cos θ
We have r² = 3rcos θ = 3x
So x² + y² = 3x
So taking derivatives:
2x + 2y dy/dx = 3
dy/dx = (3-2x)/2y

At θ = 120, r = -3/2, r² = 9/4, x = r²/3 = 3/4,
and y = rsinθ = -3√3/4
so dy/dx = (3-3/2)/(-3√3/2) = -1/√3
That is, tan(B) = -1/√3 = tan(-30°)
Since we're in the 4th quadrant, A = -60° (negative r is projected through the origin)

So B-A = -30° + 60° = 30°

Answer 2

m = tan 120° = - √3 (gradient of radius vector)
Gradient of tangent line = 1 /√3
Tangent line makes angle of 30° with x axis.

Answer 3

use the formula angle between vectors.