Question 283285: My question is about functions of standard angles.
The question is;
Find sin(-5pi/3)cos(2pi)tan(-13pi/3)cot(-29pi/6)
I am not sure how to work this out. I have tried breaking down the question to find that sin(-5pi/3) = (sqrt3)/2, cos(2pi) = 0, tan(-13pi/3)= sqrt3 but I can't find what cot(-29pi/6) is equal to.
In fact, I'm not even sure if the values I obtained for the other three were correct.
So could you please help me solve this problem.
Thanks,
Alex
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! Find sin(-5pi/3)cos(2pi)tan(-13pi/3)cot(-29pi/6)
It looks like all of these revolve around a 30/60/90 triangle.
sin(30) = 1/2
cos(30) = sqrt(3)/2
tan(30) = 1/sqrt(3)
cot(30) = sqrt(3)
sin(60) = sqrt(3)/2
cos(60) = 1/2
tan(60) = sqrt(3)
cot(60) = 1/sqrt(3)
Since it's easier to look at these in degrees, then you can convert radians to degrees by using the following formula:
degrees = radians * pi * 180 / pi.
Your original expression is:
sin(-5pi/3)cos(2pi)tan(-13pi/3)cot(-29pi/6)
-5pi/3 * 180/pi = -5*180/3 = -300 degrees
2pi * 180/pi = 360 degrees
-13pi/3 * 180/pi = -13*180/3 = -780 degrees
-29pi/6 * 180/pi = -29*180/6 = -870 degrees
If you add 360 to -300 degrees, you get an equivalent angle of 60 degrees that is in the first quadrant.
If you subtract 360 from 360 degrees, you get an equivalent angle of 0 degrees that is in the first quadrant.
If you add 2 * 360 degrees to -780 degrees, you get an equivalent angle off -60 degrees which has a reference angle of 60 degrees that is in the fourth quadrant.
If you add 3 * 360 to -870 degrees, you get an angle of 210 degrees which has a reference angle of 30 degrees that is in the third quadrant.
your original expression of:
sin(-5pi/3)cos(2pi)tan(-13pi/3)cot(-29pi/6) becomes:
sin(60)cos(0)tan(-60)cot(210)
sin(60) = sqrt(3)/2)
cos(0) = 1
tan(-60) = -tan(60) = -sqrt(3)
cot(210) = cot(30) = sqrt(3)
Now all you have to do is multiply them together.
You can also verify with your calculator.
Using the calculator in radian mode, I got the following:
sin(-5pi/3) = .866025404 which is the same as sqrt(3)/2.
cos(2pi) = 1.
tan(-13pi/3) = -1.732050808 which is the same as -sqrt(3).
cot(-29pi/6) = 1.732050808 which is the same as sqrt(3).
This agrees with the answers I got in degree mode so it's looking good.
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