Wize University Calculus 3 Textbook > Appendix: Review Table for lines and planes

The unit circle shows the values of functions sin and cos for different angles. For example we have:
cos⁡π3=12, sin⁡π3=32\cos\dfrac{\pi}{3}=\dfrac{1}{2},\ \sin\dfrac{\pi}{3}=\dfrac{\sqrt{3}}{2}cos3π​=21​, sin3π​=23​​
Note that adding or subtracting multiples of 2π2\pi2π does not change the values of trigonometric functions.
If we are looking for an angle xxx that its cos⁡\coscos equals to 0.5, we have two answers:
x=cos⁡−112⇒x=π3, x=2π−π3=5π3x=\cos^{-1}\dfrac{1}{2}\Rightarrow x=\dfrac{\pi}{3},\ x=2\pi-\dfrac{\pi}{3}=\dfrac{5\pi}{3}x=cos−121​⇒x=3π​, x=2π−3π​=35π​
In the general form we have:
x=π3+2πn, (2π−π3)+2πnx=\dfrac{\pi}{3}+2\pi n,\ \Big(2\pi-\dfrac{\pi}{3}\Big)+2\pi nx=3π​+2πn, (2π−3π​)+2πn
If we are looking for an angle xxx that its sin⁡\sinsin equals to 0.5, we have two answers:
x=sin⁡−112⇒x=π6, x=π−π6=5π3x=\sin^{-1}\dfrac{1}{2}\Rightarrow x=\dfrac{\pi}{6},\ x=\pi-\dfrac{\pi}{6}=\dfrac{5\pi}{3}x=sin−121​⇒x=6π​, x=π−6π​=35π​
In the general form we have:
x=π6+2πn, (π−π6)+2πnx=\dfrac{\pi}{6}+2\pi n,\ \Big(\pi-\dfrac{\pi}{6}\Big)+2\pi nx=6π​+2πn, (π−6π​)+2πn
An easy way to visualize and find the answers to the inverse of trigonometric functions is to draw a line from the axis and find the intersections with the unit circle.