I need to find the values of this trig functions for the negative angles; help!?

Question

I need to make sure I have the right answers for the values of these trig functions. Help?

I'm supposed to find the values of the trig functions for the negative angles, and rationalize the denominators.

Can you help please?

Thanks!


1. cos ( - 150 degrees)

2. cot ( - 330 degrees)

3. sin ( - 135 degrees)

Make sure to remember that they're negative. Thanks so much!

Answer 1

do you know about All Silver Tea Cups ?


Quadrant 1 (All) --> all trig functions are positive
Quadrant 2 (Silver) ---> sine and consine are positive
Quadrant 3 (Tea) ---> Tangent and cotangent are positive
Quadrant 4 (Cups) ---> cosec and sec are positive

break angle in terms of
n*pi/2 - teta ..
where n is an integer..

if n is odd...
sine becomes cos... cos become sine..
tan becomes cot.. cot becomes tan..
sec becomes csc.. and csc becomes sec

if n is even.. it stays the same


and based on which quadrant the angle falls in... the value is either positive or negative (based on our "all silver tea cups" deal which i mentioned above)


sine (-135)

Now.. -135 is same as (360 - 135)

so we have
sine (360 - 135) = sine (4*pi/2 - 135) = sine (4*pi/2 - pi/2 - 45)
= sine (3pi/2 - 45) = sine (n*pi/2 - 45)
n = 3 (odd) .. so sine becomes cos..

and 3pi/2 - 45 lies in.... 3rd quadrant.. and sine in 3rd quadrant (according to our "all silver tea cups") is negative

so we have
sine (n*pi/2 - 45) = - cos(45) = - 1/sqrt(2)

Answer 2

-150 degrees is the same as +210 degrees. If you go 1/2 way around a 360 degree circle you go 180 degrees -- doesn't matter which way you get there (+ or -) it is still +180 or -180. So to go -150, you are 30 degrees short of -180, or 30 degrees ahead of +180, which is 210.
Also... 360-150 = 210

-330 is the same as 30 (360-30)

-135 is the same as 225

Now, you can figure the answer based on positive angles, which you have probably done, before.

.

Answer 3

cos is an even function , meaning
cos(-x) = cos(x)
cos(-150) = cos(150) = (-1/2)*sqrt(3)

cot(-330) = sqrt(3)

sin(-135) = -sin(135) =(-1/2)*sqrt(2)