1. Reference angle for -280 degrees?

I do not understand why you add 360????

2. Theta=? If reference angle =60 and theta terminates in quadrant 3.

What does terminate mean?
Please show work and explain why.
My priority is to understand the concept ...

3. Exact value for sec theta if theta=-150 degrees
Please explain and show work

First step, add/subtract multiples of 180° until you have an angle between -180° and 180°

If θ is in QI, you're done.

If θ is in QIII, reflect through the origin. That is, add 180°. Done.

If θ is in QIV, y<0, so reflect through the x-axis. That is, add 180°.

Then, if θ is in QII, reflect it in the y-axis so x>0. That is, subtract θ from 180°.

SO, for -280°,
add 180: -100°
Now θ is in QIII, so 1dd 180: 80°

The rules get messy, and there are lots of ways to express things. Study your text, google reference angle on the web, until you find a method that makes sense. You want to end up in QI for the reference angle.

#2 Reference angle is 60, but θ is in QII, where x<0 and y<0. So rotate θ through 180°, so it winds up as 240°.

#3
reference angle for -150° is 30°.
So, since -150° is in QIII, x<0 and y<0.
secθ = r/x = 1/(-√3/2) = -2/√3

1. To find the reference angle for -280 degrees, you add 360 degrees to the given angle until you get a positive angle within the range of 0 to 360 degrees. Here's how you can do it:

-280 degrees + 360 degrees = 80 degrees

Therefore, the reference angle for -280 degrees is 80 degrees. Adding 360 degrees is necessary in order to get a positive angle within the acceptable range.

2. In trigonometry, the term "terminate" refers to where the angle ends or lies. A quadrant is a section of the coordinate plane, and there are four quadrants numbered counterclockwise from 1 to 4. Quadrant 3 is the bottom left quadrant.

To find the value of theta when the reference angle is 60 degrees and it terminates in quadrant 3, you need to determine the actual angle. Since the reference angle is positive and terminates in quadrant 3, the actual angle will be negative. Here's the explanation step by step:

1. Start with the reference angle: 60 degrees.
2. Since it terminates in quadrant 3, which is a negative quadrant, the angle will be negative.
3. So, theta = -60 degrees.

The negative sign indicates that the angle lies in quadrant 3.

3. To find the exact value for sec theta when theta is -150 degrees, you need to determine the value of sec(-150).

Here's how you can do it:

1. Start by finding the cosine of theta: cos(-150).
- Cosine is positive in Quadrants 1 and 4, so we need to find the reference angle for -150 degrees.
- To find the reference angle, add 360 degrees until you get a positive angle within the range of 0 to 360 degrees:
-150 degrees + 360 degrees = 210 degrees
- The reference angle for -150 degrees is 210 degrees.
2. Now find the cosine of the reference angle: cos(210).
- Cosine is the reciprocal of secant, so you can use the property: sec(theta) = 1/cos(theta).
- Calculate the cosine of 210 degrees using a calculator or reference table.
- cos(210) ≈ -0.342.
3. Finally, take the reciprocal of the cosine: sec(-150) = 1 / cos(210) ≈ -1 / 0.342.
- Simplifying further gives: sec(-150) ≈ -2.919.

Therefore, the exact value for sec(-150 degrees) is approximately -2.919.