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Rachel G.

Experienced High School Mathematics Teacher

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Pre-Calculus

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Question:

Why is the limit of 1/x as x approaches infinity zero?

Rachel G.

Answer:

If you think of 1/x as taking 1 thing, like a candy bar, and dividing it by a certain number of people (for now we will call that number x) this limit becomes quite concrete. If you have 2 people to split 1 candy bar, each will get 1/2. If that number gets a little bit bigger, like to 3, each person will get 1/3 of a candy bar, or a little bit less. As you keep dividing that candy bar among more and more people the amount of candy each person will get will keep getting smaller and smaller. So as x gets bigger, 1/x will keep getting smaller. In other words it will approach zero.

Trigonometry

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Question:

Why does the sine function end up making a graph that looks like a wave?

Rachel G.

Answer:

Sine is defined as the ratio of the y coordinate over the radius. Since the radius is 1 on the unit circle, we only need to look at the heights. If we think about going around that unit circle starting at the point (1,0) for the first 90 degrees the y coordinates get bigger. So the sine values would increase. Then at 90 degrees, the sine value is 1. Then the y-coordinates get smaller from 90-180 degrees so the sine curve would decrease. Then once you go from 180-270 degrees the sine values are negative so the sine curve would continue to decrease to a height of -1 at 270 degrees. Then from 270-360 degrees the y-coordinates are still negative so you still have negative sine values but they are increasing and getting closer to zero. Then after rotating 360 degrees you are back where you started at (1,0) so the wave will start over again.

Geometry

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Question:

What is a radian and how is it different from a degree?

Rachel G.

Answer:

A radian can be thought of a couple of different ways. Often it is described as the amount of rotation to move 1 radius along the edge of the circle. So if you have a circle with a radius of 3 inches and you moved 3 inches along the edge of that circle you would create an angle of 1 radian. The number of radians can also be thought of as the length of the arc divided by the radius. So if you have an arc of a circle with a radius of 3 inches that is 5 inches long, the angle creating that arc is 5/3 radians. Degrees are another unit of measure for angles that many Geometry students are more comfortable with. There are 360 degrees in a full circle while they are 2pi radians in a circle. You can convert from degrees to radians by multiplying by pi/180 and you can go from radians to degrees by multiplying by 180/pi. This ratio works because there are pi radians in 180 degrees.

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