Question 293479: Find cosx/2, given cot x=(-3),
with sign of pie/2 < x < sign of pie
Here is what I tried..
Divided the sign of pie/2 and sign of pie by 2
getting the sign of pie/4 and the sign of pie/2
which meands the sign of pies/4
Then I used the formula cosx/2=+or -the radical 1+cosu/2
Getting cos x/2= + or - the radical 1+(-3/2 divided by 2)
equaling + or - the radical (-1/2 divided by 2)
so I multiplied (-1/2) by 1/2
getting the radical of (-1)
I did not believe this was the right answer because this would give us a imaginary number. Which is not allowed. So would this be undefined?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! Your problem is to find cos( ).
You are given that cot(x) = (-3).
You are given that  < x < 
is the angle in radians.
Convert that to degrees to get *  = 90 degrees.
is the angle in radians.
Convert that to degrees to get * = 180 degrees.
x must be between 90 and 180 degrees.
Since cot(x) =  , this means that:
cot(x) = (-3) is the same as = (-3)
Multiply both sides of that equation by tan(x) and divide both sides of that equation by (-3) to get:
tan(x) = -
Take arc-tan(-) to get angle x = -18.43494882 degrees.
That angle is in the fourth quadrant.
In order for the angle to be between 90 and 180 degrees, it has to be in the second quadrant.
The equivalent angle in the second quadrant, that has the same tangent, would be -18.43494882 degrees + 180 degrees = 161.5650512 degrees.
Use your calculator to confirm that tan(-18.43494882) is the same as tan(161.5650512).
cos() = cos( ) = cos(80.78252559) = .160182243
The cosine is positive, as it should be, since the angle divided by 2 is in quadrant 1.
Your answer should be:
x = 161.5650512 degrees
cos() = .16082243
Once you knew the value of x, you could also have used the cos() formula of:
cos() = +/- 
With x = 161.5650512 degrees, you would get:
cos() = +/- 
Simplify this to get:
cos() = +/- .160182243
Since = 80.78252559 degrees, which is in quadrant 1, then the cosine of that angle has to be positive, so your answer is:
x = 161.5650512 degrees
cos() = .160182243
SOLVING THIS PROBLEM WITHOUT USING A CALCULATOR
To solve this problem without using a calculator, you would do the following.
You are given that cot(x) = -3.
Since cot(x) = adjacent / opposite, this means that either:
adjacent = -3 and opposite = 1, or:
adjacent = 3 and opposite = -1.
If adjacent = -3 and opposite = 1, this means that the angle is in quadrant 2.
If adjacent = 3 and opposite = -1, this means that the angle is in quadrant 4.
Since the angle has to be between and , this means that the angle has to be in quadrant 2 because is equivalent to 90 degrees and is equivalent to 180 degrees.
So:
adjacent = -3 and opposite = 1 putting the angle in quadrant 2.
You can solve for hypotenuse by using the Pythagorean formula of  to get  which makes 
You know that cosine of the angle is therefore equal to adjacent / hypotenuse which equals  .
You can now use the half angle formula to get:
 = +/- 
Since  , this formula becomes:
= +/-  which becomes:
= +/- 
The answer has to be positive because, if x is in quadrant 2, then has to be in quadrant 1, and cosine function in quadrant 1 is positive.
This makes your answer:
= +
You can confirm using your calculator.
 = -.948683298
 = 1.414213562
= + becomes:
= +  which becomes:
= +  which becomes:
= +  which becomes:
= + 
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