Tangent, Horizontal and Vertical Lines at the Pole

In summary, the conversation discusses finding the tangent lines at the pole r = sin5o in the interval [0,pi]. The equation for the tangent line is given and the values for f(o) and f'(o) are provided. However, there is a mistake in the numerator and it is suggested to evaluate the derivative function at the points where sin(5*theta) = 0.
  • #1
IntegrateMe
217
1
Help me understand exactly what is going on here. I'll put up an attempt at my solution:

Find the tangent lines at the pole r = sin5o, [0,pi] (note: o represents theta)

Equation: dy/dx = [f'(o)sino + f(o)coso]/[f'(o)coso - f(o)sino]

f(o) = sin5o
f'(o) = 5cos5o

plugging everything in we eventuate at:

dy/dx = [5*sino*coso + coso*sin5o]/[5cos5o*coso - sin5o*sino]

I'm stuck at this point. Any clue as to how i can advance?
 
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  • #2
IntegrateMe said:
Help me understand exactly what is going on here. I'll put up an attempt at my solution:

Find the tangent lines at the pole r = sin5o, [0,pi] (note: o represents theta)

Equation: dy/dx = [f'(o)sino + f(o)coso]/[f'(o)coso - f(o)sino]

f(o) = sin5o
f'(o) = 5cos5o

plugging everything in we eventuate at:

dy/dx = [5*sino*coso + coso*sin5o]/[5cos5o*coso - sin5o*sino]

I'm stuck at this point. Any clue as to how i can advance?
You have a mistake in the numerator below. What is f'(o)sin(o)?
dy/dx = [5*sino*coso + coso*sin5o]/[5cos5o*coso - sin5o*sino]

For which values of theta is sin(5*theta) = 0? Evaluate your derivative function at those places.
 

FAQ: Tangent, Horizontal and Vertical Lines at the Pole

What is a tangent line at the pole and how is it different from other tangent lines?

A tangent line at the pole is a line that passes through the pole of a circle or sphere and is perpendicular to the radius of the circle or sphere at that point. This is different from other tangent lines because it is the only line that can be drawn through the pole and still be tangent to the circle or sphere.

How do you determine if a line is horizontal at the pole?

A line is horizontal at the pole if it is parallel to the equator or any other line of constant latitude. This means that it does not intersect with any meridians, or lines of longitude, and is perpendicular to the radius at the pole.

Can a line be both horizontal and vertical at the pole?

Yes, a line can be both horizontal and vertical at the pole. This occurs when the line is tangent to the pole and is also parallel to the equator or any other line of constant latitude. In this case, the line is both perpendicular to the radius and parallel to the equator, making it both horizontal and vertical at the pole.

What is the significance of horizontal and vertical lines at the pole in geography or cartography?

Horizontal and vertical lines at the pole are important in geography and cartography because they help define the orientation and direction of features on a map or globe. They also play a role in determining distances and angles between different points on the Earth's surface.

Can a line be tangent to the pole at more than one point?

No, a line can only be tangent to the pole at one point. This is because the pole is a single point on a circle or sphere, and there can only be one tangent line through that point. If a line appears to be tangent to the pole at multiple points, it is actually tangent to a different circle or sphere with a smaller radius that intersects with the original circle or sphere at those points.

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