SOLUTION: FInd the values of θ between 0° and 180° for which a) secθ= cosecθ b) tanθ= -1 c) sinθ - sqrt(3) cosθ = 0

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Question 1125743: FInd the values of θ between 0° and 180° for which
a) secθ= cosecθ
b) tanθ= -1
c) sinθ - sqrt(3) cosθ = 0
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
it helps to put all this stuff in the first quadrant and then figure out what quadrant it really belongs in.
so far, that's worked ok for me.

i'll use x instead of theta.
anwswer will be the same even though the variable name used is different.
theta means the same as x.
using x makes graphing a little easier.

problem A.

start with sec(x) = csc(x)
sec(x) is equal to 1/cos(x)
csc(x) is equal to 1/sin(x)
therefore 1/sin(x) = 1/cos(x)
cross multiply to get xin(x) = cos(x)
divide both sides of this equation by cos(x) to get:
sin(x)/cos(x) = 1
sin(x)/cos(x) = tan(x)
therefore tan(x) = 1
that makes x = arctan(1) = 45 degrees.
that's in the first quadrant.
tangent is positive in the first quadrant and the third quadrant.
0 to 180 degrees is in the first quadrant and the second quadrant.
therefore x = 45 degrees in the interval between 0 and 180 degrees.
here's the graph of y = sec(x) and the graph fo y = csc(x).
their intersection is equal to 45 degrees in the interval between x = 0 and x = 180 degrees, confirming the solution is correct.
the following graph confirms the solution is correct.
$$$

problem B.

start with tan(x) = -1
place it in the first quadrant by making it positive.
you get tan(x) = 1.
this makes x = arctan(1) which makes x = 45 degrees.
your angle is 45 degrees in the first quadrant.
but you want tan(x) to be equal to -1.
tan(x) is positive in the first and third quadrant, but negative in the second and fourth quadrant.
therefore, your angle has to be in the second quadrant.
the equivalent angle in the second quadrant is equal to 180 - 1 = 135 degrees.
the angle that gives you tan(x) = -1 in the interval between 0 and 180 degrees is therefore 135 degrees.
the following graph confirms the solution is correct.
$$$

problem C.

start with sin(x) - sqrt(3) * cos(x) = 0
add sqrt(3) * cos(x) to both sides of the equation to get:
sin(x) = sqrt(3) * cos(x)
divide both sides of this equation by cos(x) toget:
sin(x) / cos(x) = sqrt(3)
sin(x) / cos(x) = tan(x)
equation becomes tan(x) = sqrt(3).
this makes x = arctan(srt(3) which makes x = 60 degrees.
tan(x) is positive in the first and third quadrants, therefore:
the solution is x = 60 degrees in the interval between 0 and 180 degrees.
the following graph confirms the solution is correct.
$$$