Question 1107920: Find the values of the trigonometric functions of t from the given information.
csc(t) = 9, cos(t) < 0
Found 2 solutions by Theo, greenestamps:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! csc(t) = 9
since csc = 1/sin, you get 1/sin(t) = 9.
solve for sin(t) to get sin(t) = 1/9.
solve for t to get t = arcsin(1/9) = 6.379360208 degrees.
cos(t) is negative in the second and third quadrants.
sin(t) is positive in the first and second quadrants.
therefore, your angle has to be in second quadrant because cosine is negative and sine is positive in the second quadrant.
note that sine and cosecant, being reciprocals of one another, follow the same sign in their respective quadrant.
same goes for cosine and secant, and for tangent and cotangent.
what this means to your problem is that, if sine is positive in first and second quadrant, so is cosecant.
the only reason to use the fact that cosecant equals 1/sine is because most calculators only do sine and tangent and cosine.
you have to convert cosecant and cotangent and secant to 1/sine and 1/tangent and 1/secant in order to find their trigonometric functions using those calculators.
that's why i converted csc(t) = 9 to 1/tan(t) = 9 and then solved for tan(t) = 1/9.
so, your angle is in the second quadrant.
the angle in the first quadrant is 6.379370208.
the angle in the second quadrant is 180 minus that = 173.6206298.
you can use your calculator to find cos(173.6206298) and csc(173.6206298).
since cosecant is reciprocal of sine, then find 1/sin(173.6206298) to get csc(173.6206298).
you get:
cos(173.6206298) = -.99380799, which is negative.
1/sin(173.6206298) = 9, which is positive and is the same as csc(t) = 9 that we started with.
the equivalent angle in the second quadrant will have the same trigonometric function as the equivalent angle in the first quadrant, except for the sign, which may or may not be the same.
the basic trig functions and their signs are shown below:
quadrant 1 2 3 4
sine + + - -
cosine + - - +
tangent + - + -
if you're looking for sine to be positive and cosine to be negative, then you're in quadrant 2 only.
your solution is that t is equal to 173.6206298 degrees.
Answer by greenestamps(13203)  (Show Source):
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