If cos x=8/17, and 3π/2<x<2π, what is sin(x−π)?

Question

1) If tan ⁡x=60/11, and π<x<3π/2, what is sin⁡(x−π)?

2) If cos x=8/17, and 3π/2<x<2π, what is sin(x−π)?

Answer 1

1) Draw a coordinate plane with a circle about the origin. The directions say to work in quadrant 3 (that is where the angle (x) is greater than π and less than 3π/2. The draw a point on the circle in Q3 and make a right triangle that has one vertex at the origin, one at the point on the circle you drew, and one on the x-axis. Think of the reference angle as "x". Since tan(x) = opposite/adjacent = 60/11, call the opposite side 60 and the adjacent side 11. Like this:

Use the Pythagorean Theorem to solve for the hypotenuse:

hyp = √(11² + 60²) = 61

Now, consider that x - π is the extension of the hypotenuse back in quadrant 1 so it's the same triangle flipped around in Q1 like this:

So the sin(x- π) = opp/hyp = 60/61

Do the same for problem 2 except you'll be in Q4 and when you flip the triangle, you'll be in Q2.