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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 -------------------------- 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 ------------------------- c² = x² + y² c² = 4² + 3² c² = 25 c = √25 c = 5 ------------------------- 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 ------------------------- 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 ------------------------ 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 ----------------------- 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 ---------------------- 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 --------------------- see attachment