Course: Trigonometry
Topic: The Six Trigonometric Functions
Subtopic: xyr Definition


This lesson marks the *real* beginning of trig by introducing the six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. While functions like `f(x)=x^3` have a variable input and an expression output, the trig functions, such as `f(theta)=sin(theta)`, have an angle input and a ratio output. The output is the ratio of two sides of a right triangle where the angle, `theta` pronounced "theta", is an interior angle of the triangle.

We will be defining these functions by expressing the ratio outputs in two different, but equivalent, ways. The first definition we use is the xyr definition meaning that the six trig functions have outputs in terms of the sides of a right triangle, x, y, and r, where r is the hypotenuse.


By the end of this topic you should know and be prepared to be tested on:


Memorize the xyr definition of the six trig functions:

Triangle with labels x, y, r. Definitions cosine of theta equals x over r, sine of theta equals y over r, tangent of theta equals y over x, and their six reciprocals cosecant, secant, and cotangent, respectively.

Memorize (or be able to quickly determine) the sign of trig functions in each quadrant:

As described all positive in quadrant I, sine positive in quadrant II, etc. "All Students Take Calculus" can be used to quickly determine which quadrants produce positive answers. For example if the angle input is in QIII, Tangent (and its reciprocal cotangent) are the only functions that have positive answers.

Be able to quickly determine the value of the trig functions at each quadrantal angle. Be sure you understand how to find the values in this chart:

Chart with five quadrantal angles (0, 90, 180, 270, 360 degrees), and six trig functions at each being either 0, -1, 1, or undefined.

Mini-Lectures and Examples

STUDY: Angles, Triangles, and the xyr Definition of Trig Functions

Supplemental Resources (optional)

Videos from James Sousa's MathIsPower4U:

Mini-Lesson:Introduction to Trigonometric Function Using Angles in Standard Position

Determine Trig Function Values Given a Point on the Terminal Side of an Angle
Example:  Determine Trig Function Values from Given Information

rev. 2020-09-12