Unit Circle Definition of Trig Functions

The Unit Circle  A unit circle is the circle with center at the origin and radius equal to 1 (one unit).  Its equation is (x – 0) 2 + (y – 0) 2 = 1 or x 2 + y 2 = 1.

Coordinate Function

 0, π/2, π, 3π/2, and 2π (and their multiples) are called quadrantal angles.  Note that P(θ) = (x, y).

Circular Functions  Trigonometric functions are defined so that their domains are sets of angles and their ranges are sets of real numbers. Circular functions are defined such that their domains are sets of numbers that correspond to the measures (in radian units) of the angles of analogous trigonometric functions.

Circular Functions  The ranges of these circular functions, like their analogous trigonometric functions, are sets of real numbers.  These functions are called circular functions because radian measures of angles are determined by the lengths of arcs of circles. In particular, trigonometric functions defined using the unit circle lead directly to these circular functions.

Circular Functions  The circle below is drawn in a coordinate system where the circle's center is at the origin and has a radius of 1. This circle is known as a unit circle.

Circular Functions  Definition: the sine and the cosine functions in the set of real numbers are defined by x = cosine θ and y = sine θ, where θ R and P(θ) = (x, y).  The x and y coordinates for each point along the circle may be ascertained by reading off the values on the x and y axes. If you picture a right triangle with one side along the x-axis:

Signs of the Circular Functions Note: r is always positive. Thus, the signs of the values of the trigonometric functions of angle θ are determined by the signs of x and y.

 The signs of the trigonometric functions depend upon the quadrant in which the terminal side of θ lies. A S T C - All Students Take Calculus! A S T C - All Silly Trig Classes!

Some Fundamental Circular Functions Identities

Examples

Special Angles

330°

Other illustrations…

Quadrantal Angles  An angle in standard position whose terminal side lies on an axis (either the x-axis or the y- axis) is said to be quadrantal.

Quadrantal Angles Example: Find the circular functions of 2π.

The Unit Circle

Reference Angles… (a recall)

Evaluating Trigonometric Functions Using Reference Angles Procedures: 1. Determine the reference angle θ’ associated with θ. 2. Find the value of the corresponding trigonometric function of θ’. Exact value can be obtained if θ’ is 30°, 45°, or 60°, or it can be an approximate value from a calculator or a table. 3. Affix the proper algebraic sign for the particular function by determining the signs of P(x, y) where the terminal side of θ lies.

Examples Use reference angles to solve the following. 1. Find tan 120°. 2. Approximate the value of cos (-16°48’) by first expressing it in terms of a function of the associated reference angle. 3. Approximate the value of sin 220°.

Exercises

Do Worksheet 5