question form

Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.

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@& -pi / 3 coincides with -60 degrees, which coincides with 300 degrees.

The coordinates of the -pi/3 point on a circle of radius 1 would be (1/2, -sqrt(3) / 2).*@

@& If you know the coordinates of the 30, 45 and 60 degree points, then you can easily find the coordinates the 120, 135, 150, 210, 225, 240, 300, 315 and 330 degree points, all of which lie at either 30, 60 or 45 degrees from the x axis, but in different quadrants.*@

@& Exactly.*@

@& If the angle is negative you go in the clockwise direction around the circle, starting as always from the positive x axis.

So -270 degrees would be coterminal ('coterminal' means 'ending at the same point') with +90 degrees.*@

@& Right so far.*@

@& Locate 120 deg on the unit circle, note that this angle is at 60 degrees with the negative x axis, and use what you know about the special angles to find the coordinates of the point.*@

@& Locate 150 deg on the unit circle, note its angle with the x axis, and write down the coordinates of the point. Use the coordinates to find the tangent.*@

@& First see my note below.

Then identify the cosine as an even or an odd function.

How therefore is cos(-30 deg) related to cos(30 deg)?

What is cos(30 deg), and what therefore is cos(-30 deg).*@

@& An even function has the property that f(-theta) = f(theta).

An odd function has the property that f(-theta) = - f(theta).

These are standard properties studied in first-semester precalculus courses; check your text and see if they are covered, and if necessary review them. If they aren't covered (or if after working through this you aren't sure you understand) let me know.

Consider theta = 30 degrees. This corresponds to a positive rotation of 30 degrees, starting from the positive x axis.

cos(theta) is the x coordinate of the unit-circle point.

Now locate -theta = - 30 deg on the unit circle. What is the x coordinate of this point? How does it compare with the x coordinate of theta = 30 deg?

You should have seen the x coordinates are the same.

Now continue both rotations until theta = 60 degrees, with -theta = -60 degrees. How do the x coordinates compare?

You can continue this process to any value of theta. If your positive and negative rotations are equal and opposite, your x coordinates will always be the same.

Now look at the y coordinates. You should see that they are all equal and opposite (e.g., the y coordinate for 30 deg is positive, the y coordinate for -30 deg is negative, and by symmetry the two are equal and opposite). The same is true for 60 deg and -60 deg, or for any other angle theta.*@

@& So the cosine function is even. cos(theta) = cos(-theta).

The sine function is odd. sin(theta) = -sin(-theta).

Now what does this imply about csc(theta)? What is implied about sec(theta)?

*@

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@& Let me know if my notes don't help, or if you have additional questions.*@