The trigonometric functions are periodic wave functions that are used throughout math and physics. This page introduces and defines the functions using the geometry of the right triangle. See this page the circle definitions of the functions.
The functions are defined below using the sides of the right triangle. Each function takes an angle of a right triangle as input and returns the ratio of two of its sides.
The functions sine, cosine and tangent are the most commonly used trigonometric functions. This page uses the degree angle system for measuring angles on the right triangle, because degrees are often used to introduce the functions. However, in practice, the functions use the radian angle system as the default unit^{[1]}.
Given an angle of a right triangle the sine function returns the ratio of the opposite side over the hypotenuse of the triangle. For example, given an angle of the function returns the ratio . This is shown below.
Another example of a common trigonometric ratio is given by the 454590 right triangle shown below. To find the ratio, substitute the opposite and hypotenuse lengths into the definition.
Note, mathematicians don’t like having the square root operator in the denominator so they often rewrite the ratio by multiplying the expression by one in the form . This transformation is shown below.
Given an angle of a right triangle the cosine function returns the ratio of the adjacent side over the hypotenuse of the triangle. For example, given an angle of the function returns the ratio . This is shown below.
Another example of a common trigonometric ratio is given by the 454590 right triangle shown below. To find the cosine ratio, substitute the opposite and hypotenuse lengths into the definition.
Note, as discussed in the sine section mathematicians don’t like having the square root operator in the denominator so they often rewrite the ratio as shown below.
Given an angle of a right triangle, the tangent function returns the ratio of the opposite side over its adjacent side. For example, given the angle of the tangent function returns the ratio of .
As discussed in the sine section, mathematicians don’t like having the square root operator in the denominator so they often rewrite the ratio by multiplying by one as .
To convert an angle from radians to degrees multiply by the ratio of 360° divided by τ (tau) radians.
There are six trigonometric functions that relate to the geometry of the righttriangle sine, cosine, tangent, cosecant, secant, and cotangent. The functions take the angle of a right triangle as input and return a ratio of two of its sides.
The sine function returns the sine of a number provided in radians. In geometric terms, the function returns the vertical component of the point formed by the angle on the unit circle.
The cosine function returns the cosine of an angle provided in radians. In geometric terms, the function returns the ratio of the righttriangle's adjacent side over its hypotenuse.
Given the angle of a right triangle as input returns the ratio of the triangle's opposite side over its adjacent side.
Degrees are a unit of measure for angles. A full rotation is equal to 360 degrees. In the XY Cartesian Coordinate System, degrees are measured starting from the rightmost edge of the circle.
Radians are a unit that measure angle using the radius of a circle. One radian is equal to the amount of rotation required to travel the length of one radius along the circumference of the circle.

Radians Versus DegreesWumbo (internal)