Trigonometric Functions Introduction

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.

Introduction

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 right triangle has three sides relative to the angle that defines its shape: the adjacent, opposite and hypotenuse.
Figure 1: Right Triangle

Functions

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].

Note: If you want to use degrees to calculate a trigonometric ratio, make sure that your calculator is set to degrees. Otherwise, convert the angle to radians.

Sine

The right triangle has three sides relative to the angle that defines its shape: the adjacent, opposite and hypotenuse.
Figure 2: Right Triangle

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.

The 30 60 90 special right triangle is defined by the angles: 30, 60 and 90 degrees.
Figure 3: 30 60 90 Triangle

Another example of a common trigonometric ratio is given by the 45-45-90 right triangle shown below. To find the ratio, substitute the opposite and hypotenuse lengths into the definition.

The 45 45 90 special right triangle is defined by the three angles: 45, 45 and 90 degrees. Its simple geometry makes it useful in applications.
Figure 4: 45 45 90 Triangle

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.

Cosine

The right triangle has three sides relative to the angle that defines its shape: the adjacent, opposite and hypotenuse.
Figure 5: Right Triangle

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.

The 30 60 90 special right triangle is defined by the angles: 30, 60 and 90 degrees.
Figure 6: 30 60 90 Triangle

Another example of a common trigonometric ratio is given by the 45-45-90 right triangle shown below. To find the cosine ratio, substitute the opposite and hypotenuse lengths into the definition.

The 45 45 90 special right triangle is defined by the three angles: 45, 45 and 90 degrees. Its simple geometry makes it useful in applications.
Figure 7: 45 45 90 Triangle

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.

Tangent

The right triangle has three sides relative to the angle that defines its shape: the adjacent, opposite and hypotenuse.
Figure 8: Right Triangle

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 .

The 30 60 90 special right triangle is defined by the angles: 30, 60 and 90 degrees.
Figure 9: 30 60 90 Triangle

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 .

Links

References

  1. Radians Versus Degrees
    Wumbo (internal)