Find sin θ and cos θ

Question

if the terminal side of θ lies along the line y = −2x in Quadrant IV.

Answer 1

With the given equation, the slope is -2 and the y-intercept is 0. Since we are in the 4th quadrant, the line will have to pass through the point ( 1 , -2 ) to maintain this slope. If we form a right triangle using this portion of the line in the 4th quadrant as the hypotenuse, then this makes the vertical leg of the right triangle to be 2 units in length and the horizontal leg to be 1 unit in length. ** Note: the vertical leg would extend from the line up to the positive x-axis and the horizontal leg would run to the right along the x-axis, while θ will be our angle rotated from the positive x-axis, counter-clockwise, around to our line in the 4th quadrant.

By Pythagorean Theorem, we can determine that the hypotenuse of our right triangle would be √5 units long. Now we can write the sine and cosine ratios for our triangle with sin θ = -2/√5 (or, if we rationalize the denominator, -2√5/5) and cos θ = 1/√5 ( or √5/5).