# What are what are sin tan and cos and how do they relate to the Pythagorean theorem?

embizze | Certified Educator

sin, cos, and tan can be considered as trigonometric functions or as ratios of sides of a right triangle. We will use the second interpretation.

In a right triangle, consider one of the acute angles; let it be A. The other acute angle will be B and the right angle C with the sides opposite the angles labelled a,b, and c respectively.

The sin (sine) is defined as the ratio of the side opposite the angle to the hypotenuse; so`sinA=a/c`

The cos (cosine) is defined as the ratio of the side adjacent the angle to the hypotenuse; so`cosA=b/c`

The tan (tangent) is defined as the ratio of the leg opposite the angle to the leg adjacent the angle; so`tanA=a/b`

The Pythagorean theorem says that for this right triangle `a^2+b^2=c^2`.

If we take`sin^2A+cos^2A` we get `(a/c)^2+(b/c)^2` or`(a^2+b^2)/c^2`; using the Pythagorean theorem we can rewrite the numerator as`c^2` so that:

`sin^2A+cos^2A=1`