Expert: Josh - 11/16/2007

Question
1. q = 50 degrees

2. q = 120 degrees

3. q = 6p/7

4. q = 3.3

5. q = 300 degrees

6. q = -145 degrees

Answer
Hi Sean,

By "reference angle", I assume you mean restricting the angle to the range which extends from -90 to 90 degrees. I will assume that 0 degree points in the direction to the right and angle is measured anti-clockwise.

For Q2, 120 degrees is subtended by BCD in the sketch below. As far as trigonometry is concerned, sin(120)=sin(180-120)=sin(60), so the reference angle is 180-120=60 degrees. Take note that the line CB is in the so called "second quadrant". i.e., the angle is between 90 degrees and 180 degrees.

B




A------------C-----------D

Definition: An angle "q" is in the
a) first quadrant, if 0 <= q < 90;
b) second quadrant, if 90 <= q < 180;
c) third quadrant, if 180 <= q < 270;
d) fourth quadrant, if 270 <= q < 360;
when we restrict "q" to 360 degrees.

Note: 540 degrees, for instance, is same as 180 degrees, since 180+360=540. Effectively, we wrap around the circle once for every 360 degrees. By the same token, 45 degrees is the same as 45+360=405 degrees.

Q3. First of all, pi is equivalent to 180 degrees. Since (6/7)*pi is greater than (1/2)*pi and less than pi, it is greater than 90 degrees, but less than 180 degrees. According to our definition, it sits in the second quadrant.

When it belongs to the second quadrant, the reference angle is obtained by subtracting it from pi. i.e., pi-(6/7)*pi = (1/7)*pi.

Q4. I'm not sure if this is measured in radians or degrees. If it is measured in radian, remember that "pi" has a value close to 3.1415. To convert this to degree, you divide 3.3 by pi (this is approximately 3.1415...) and times it by 180.  In any event, because it (the 3.3) is slightly greater than pi (3.1415...) it falls in the third quadrant. i.e., it is somewhere between 180 and 270 degrees (much closer to 180 than 270). To obtain the reference angle in this case, when q is in the third quadrant, we calculate (3.3/3.1415)*180 - 180. This ensures the reference angle is between 0 and 90 degrees.

Q5, q is in the fourth quadrant. Ans: 360 - q = 60

Q6, -145 is in the third quadrant. Use a protractor or draw a rough sketch on paper to see this if you're not sure. In fact, -145 = -145+360 = 215 degrees. As we have done before, for angles in the third quadrant, we take q-180. This gives 35 degrees. The answer -35 degrees may also be acceptable depending on how you define reference angle, whether it is bounded between 0 and 90 or goes from -90 to 90. Check with your teacher on this.