I'm gonna use alpha instead of theta..
If tan α = 3/4, then we can say that y/x is 3/4
y = 3
x = 4
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If we are to draw a triangle in representation for the angle, x and y are the legs of the triangle. For us to find the hypotenuse, let us use Pythagorean Theorem.
c² = x² + y²
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c² = x² + y²
c² = 4² + 3²
c² = 25
c = √25
c = 5
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sine α = opposite/hypotenuse
or in this case,
sine α = y/c
Since y in the 3rd quadrant is negative, then take the negative of y
sin α = -3/5
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cosine α = adjacent/hypotenuse
or in this case,
cos α = x/c
Since x in the 3rd quadrant is negative, then take the negative of x
cos α = -4/5
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cotangent α = cosine/sine
cot α = (4/5)/(3/5)
5 will be cancelled out
Both x and y in the 3rd quadrant are negative and negative over negative is positive, then it will have a positive sign
cot α = 4/3 or merely the reciprocal of tangent
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secant α = 1/cosine α
sec α = 1/(4/5)
Since x in the 3rd quadrant is negative, then take the negative of x
sec α = -5/4
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cosecant = 1/sine α
csc α = 1/(3/5)
Since y in the 3rd quadrant is negative, then take the negative of y
csc α = -5/3
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see attachment
