Trigonometry Values Pi/4

For a Right Isosceles Triangle, symmetry requires that the angle at each Vertex be given by

$${\textstyle{1\over 2}}\pi+2\alpha = \pi,$$ (1)

so $\alpha = \pi/4$. The sides are equal, so

$$\sin^2\alpha+\cos^2\alpha = 2\sin^2\alpha = 1.$$ (2)

Solving,

$$\sin\left({\pi\over 4}\right) = {\textstyle{1\over 2}}\sqrt{2}$$ (3)

$$\cos\left({\pi\over 4}\right) = {\textstyle{1\over 2}}\sqrt{2}$$ (4)

$$\tan\left({\pi\over 4}\right) = 1.$$ (5)

See also Square