Exact values of trig functions.

[Jimbo57]

So basically I know what the answer is to the problem and the steps on how to get there, but during the steps I'm not sure why one thing happens.

sec(7pi/6)
=1/(cos7pi/6)
=1/(-cos(pi/6)/ <--- I'm not sure why the 7 disappears in this step and the cos becomes negative.

I get the feeling this is something I should know, but it's been a while. Any help would be appreciated.

 

[Dick]

cos(x)=(-cos(x-pi)). Look at the graph of cos or use trig sum rules. Like cos(a+b)=cos(a)cos(b)-sin(a)sin(b). Does that help?

 

[Jimbo57]

Thanks for your time Dick. So from what I understand, the 7 disappears because we want to simplify the function to get an exact value. Since it is a sine wave, there are multiple moments along the graph where the function equals the same thing, especially when there is more than one period, ie > 2pi. Am I out to lunch here or am I in the right direction?

 

[NascentOxygen]

Yes, it simplifies it. Negative one-sixth of Pi sounds simpler than seven-sixths of Pi.

You might be able to discover some of this by playing with your calculator.

Set it to RADIANS, since we are working in radian measure, not degrees. How do we know this? Because you don't see the ^o symbol anywhere.

Evaluate 7*Pi/6
Press COS

Now, let's backtrack.
You see the answer has a - sign, so change it to +
Now press INVERSE COS
The answer is more understandable (to maths people) if it is expressed in fractions of Pi, so divide it by Pi
Whatever you see now, is how many Pi it was/is. The fraction may not be recognizable, so hit 1/x to find its reciprocal. Ah hah! Whatever it was is revealed as having been 1/6.

So the COS(7Pi/6) that you started with has been found to be equal to the COS of Pi /6 provided you don't forget to reinsert that negative sign in front of the COS that we earlier rubbed out.

 

[Jimbo57]

Wow, it's too bad the text couldn't explain it like you did NO. Thanks amigo.

 

[Macarooney]

Another explanation for the -cos(pi/6) is because that's the reference angle of cos(7pi/6); the cos being negative because it lies in Quadrant III as opposed to the first or fourth quadrant, where it would be positive. The reference angle tells you what kind of right triangle you're working with, and if you remember the dimensions for each (a 30-60-90 triangle in this case), you should be able to plug that in for secant (which is 1/cos).

Hopefully I didn't make things more confusing, and I'm assuming by your last post that you figured it out, but that's just another way of looking at it. :)