Hi, I am currently tutoring yr 11 and 12.
I had difficulty explaining how to determine unit circle values without the use of known formulas
tan(3*pi/2 - a) = cot(a) OR sin(pi/2 - b) = cos(b)
How can I explain it with just the unit circle - not the given equivalencies?
using "radian" notation, we have that the sum of the 3 angles of any triangle is equal to π. if our triangle is a right triangle, then our "main angle" (the adjacent angle, formed by the hypotenuse and the x-axis) and the "other angle" (the opposite angle, formed by the hypotenuse and the vertical leg) sum to π/2.
if we "swap" the angles, we turn the vertical leg into a horizontal one, and the x-axis becomes the y-axis (imagine taking that same triangle, and moving the point that was at the center of the unit circle to the outer radius, while moving the point that was on the circle radius to the origin, and then "rotating and flipping" to switch the axes).
this means that the side that was the cosine, is now the sine of the opposite angle, and that the side that was the sine, is now the cosine of the opposite angle. so
sin(π/2 - θ) = cos(θ)
cos(π/2 - θ) = sin(θ)