Expert: Paul Klarreich - 10/14/2008

Question
Trigonometric Functions and another related question. I'm having trouble with the every part of this problem. In #1 I cannot find the angle measures which prohibits me from doing just about the rest of the problem. The same goes for #2. Any help with these problems would be greatly appreciated. Thanks again!

1) The point at (-3, -√3) lies on the terminal side of an angle in standard position.

a) Give the degree measure of 3 angles that fit the description
b) Give the radian measure of all angles that fit the description
c) Tell how to find the cosine of such angles. Give the cosine of these angles.
d) Name angles in the first, second, and third quadrants that have the same reference angle as those above.
e) Write the coordinates of a point in Quadrant II. Find the values of the six trigonometric functions of an angle in standard position with this point on its terminal side.

2) There is a circle with a radius of 10 inches. There is a point A on the farthest point to the left on the circle. (-10, 0).

a) Write an equation to describe the motion of the point A as the circle turns in place, completing a revolution once every 20 seconds.

b) How would the equation change if the radius of the wheel were 5 inches?

c) How would the equation change if the radius of the wheel were 5 inches?

d) How would the equation change if point A was at the top of the circle at t=0?  

Answer
Hi, Mike,

The keys to this lie in:

I. Making a good diagram. Really good.
II. Knowing your stuff about the basic 'special' triangles --

  A. The 45-45-90, with sides like 1,1,√2
  B. The 30-60-90, with sides like 1,√3,2
........................

1) The point at (-3, -√3) lies on the terminal side of an angle in standard position.

This is in Q3, since x,y are both neg.

        |
        |
        |
        |
S -3 = x|
-+-------O---------
|       |
|       |
|       |-√3 = y
|       |
P/       |

Use the Pythagorean Thm to find  OP = √12 = 2√3 = r

Since we are in Q3, our actual angle is theta = 180 + ref angle.

For the reference angle,  tan t = y/x = √3/3, which matches the 1,√3,2 triangle, with

√3 <=> 1;  3 <=> √3, and  2√3 <=> 2.  (Check this arithmetic and convince yourself.)

That means the angle (ref) opposite √3 is 60.

And our actual angle is 180 + 60 = 240 degrees.
............
a) Give the degree measure of 3 angles that fit the description

It could be 240 + 360 = 600, or 240 - 360 = -120, as well as 240.

.........................
b) Give the radian measure of all angles that fit the description.

Solve:

degrees    radians
------- = --------
180         pi

For ex,

 240     radians
------- = --------
180         pi

 4       radians
------- = --------
 3         pi

radians = 4pi/3

etc.

c) Tell how to find the cosine of such angles. Give the cosine of these angles.

Well, all these are COTERMINAL angles -- same diagram -- so they all have

cosine theta = x/r.


d) Name angles in the first, second, and third quadrants that have the same reference angle as those above.

If ref angle = 60, you could have:

Q1 : 60
Q2 : 120
Q3 : 240 (the one you had)
Q4 : 300.
...............
e) Write the coordinates of a point in Quadrant II. Find the values of the six trigonometric functions of an angle in standard position with this point on its terminal side.

This is unrelated, really.  Pick  x = -3 (in Q2),  y = 4. Then r = 5, and use the definitions.

...................................
2) There is a circle with a radius of 10 inches. There is a point A on the farthest point to the left on the circle. (-10, 0).

..................

a) Write an equation to describe the motion of the point A as the circle turns in place, completing a revolution once every 20 seconds.

b) How would the equation change if the radius of the wheel were 5 inches?

c) How would the equation change if the radius of the wheel were 5 inches?

d) How would the equation change if point A was at the top of the circle at t=0?

...............
a) Write an equation to describe the motion of the point A as the circle turns in place, completing a revolution once every 20 seconds.

ARE YOU ABSOLUTELY SURE THE PROBLEM SAID THIS?  ARE YOU ABSOLUTELY SURE IT DID NOT SAY:

a) Write two equations to describe the motion of the point A as the circle turns in place, completing a revolution once every 20 seconds.
................
Assuming it said that, your equations use r = 10, and, and the center is (0,0).

First write a function of t = time in seconds.

The angular speed is  1/20 rev per second.  But a rev = 2pi.

So  you want theta = (1/20)2pi t = pi t/10

HOWEVER, you are starting at  theta = pi, so you must add that:

theta = pi t/10 + pi.

x = r cos theta
y = r sin theta

where you just put  r = 10, theta = pi t/10 + pi

I think you can handle the rest.