Question 560300: Find θ for 0≤θ≤2pie in the following
Cot θ=0.4291 and cos θ<0
Please help
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! your interval is from 0 radians to 2 * pi radians which is equivalent to:
from 0 degrees to 360 degrees.
first you want to solve for cotangent theta = .4291
since cotangent theta is equal to 1 / tangent theta, your equation becomes:
1 / tangent theta = .4291
multiply both sides of this equation by cotangent theta and divide both sides of this equation by .4291 to get:
1 / .4291 = tangent theta.
simplify to get:
tangent theta = 2.331002331
to find theta, you take the arc tangent of 2.331002331 to get:
theta = 66.78066768 degrees.
that's the value of theta if theta is in the first quadrant.
in the first quadrant, tangent is positive and cosine is positive, so your answer can't be in the first quadrant because cosine theta has to be less than 0 which means cosine theta must be negative.
tangent theta is positive in the first quadrant and in the third quadrant.
cosine theta is negative in the second quadrant and in the third quadrant.
since the third quadrant allows tangent to be positive and cosine to be negative, your answer must be in the third quadrant.
the equivalent angle in the third quadrant that has the same tangent as an angle in the first quadrant would be equal to 180 + 66.78066768 degrees which makes the angle equal to 246.7806677 degrees.
the equivalent angle in radians is equal to 246.7806677 * pi / 180 which is equal to 1.371003709 * pi radians which is equivalent to 4.307135181 radians.
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if you had not converted to degrees, but worked in radians from the beginning, you would have done the following:
your interval is from 0 radians to 2 * pi radians.
first you want to solve for cotangent theta = .4291
since cotangent theta is equal to 1 / tangent theta, your equation becomes:
1 / tangent theta = .4291
multiply both sides of this equation by cotangent theta and divide both sides of this equation by .4291 to get:
1 / .4291 = tangent theta.
simplify to get:
tangent theta = 2.331002331
to find theta, you take the arc tangent of 2.331002331 to get:
theta = 1.165542528 radians
that's the value of theta if theta is in the first quadrant.
in the first quadrant, tangent is positive and cosine is positive, so your answer can't be in the first quadrant because cosine theta has to be less than 0 which means cosine theta must be negative.
tangent theta is positive in the first quadrant and in the third quadrant.
cosine theta is negative in the second quadrant and in the third quadrant.
since the third quadrant allows tangent to be positive and cosine to be negative, your answer must be in the third quadrant.
the equivalent angle in the third quadrant that has the same tangent as an angle in the first quadrant would be equal to pi + 1.165542528 radians which makes the angle equal to 4.307135181 radians.
this is equivalent to 1.371003709 * pi radians.
to make this into degrees multiply by 180 and divide by pi to get 246.7806677 degrees.
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