Hi Dorian S.,
For a right triangle, sin(θ) is equal to the side opposite of ∠θ over the hypotenuse of the right triangle (SO/H). Since we know the two sides of the right triangle we can find the third side using the Pythagorean theorem ( a2 + b2 = c2 ). Since (c) is the hypotenuse (H), and we can choose either (a) or (b) to be the side opposite (SO), we can rearrange the theorem and solve for the side adjacent (SA). So b = √( c2 - a2 ), where (b) is now our side adjacent. Lets find (b).
b = √( 52 - [-4]2 ) = 3, so our triangle is a 3-4-5 right triangle with 3 being equal to our side adjacent (SA).
The problem also states that this right triangle is in the IV quadrant. This makes our x values positive (+) and our y values negative (-). If we were to plot our triangle in quadrant IV, with theta (θ) being the angle created between the x-axis and the hypotenuse, we would see that the side adjacent corresponds to our (SA = x) values and the side opposite corresponds to our (SO = y) values. This is why (y) is negative (y = -4). We can also see that our (x) is positive (x = 3).
Next we know that cot(θ) is the side adjacent over the side opposite (SA/SO). Lets find cot(θ).
cot(θ) = SA/SO = 3/-4 or -3/4.
I hope this helps, Joe.
