sin(75)=(√6+√2)/4=0.9659 approx; cos(75)=(√6-√2)/4=0.2588.
sin(135)=√2/2=0.7071 approx; cos(135)=-√2/2=-0.7071 approx.
sin(270)=-1; cos(270)=0.
To find these from scratch note that 270º=3π/2 and that 135=270/2.
Trig identity: cos(2θ)=1-2sin²(θ)=2cos²θ-1.
Therefore: sin²θ=(1-cos(2θ))/2 and cos²θ=(1+cos(2θ))/2.
When 2θ=270, sinθ=cosθ=1/√2=√2/2. But we need to decide on whether it’s the positive or negative square root. The ASTC Rule says that sinθ=√2/2 and cosθ=-√2/2, when θ=135 (quadrant 2).
75=90-15 so sin(75)=cos²(15)=(1+cos(30))/2.
cos(30)=√3/2, so cos²(15)=(1+√3/2)/2=(2+√3)/4. It can be shown that √((2+√3)/4)=(√6+√2)/4 (square this and you will see).