Trigonometric Functions

Radian Measure

Although we can use both radians and degrees, radians are a more natural measurement because they are related directly to the unit circle, a circle with radius 1. Given an angle θ, let s be the length of the corresponding arc on the unit circle. We say the angle corresponding to the arc of length 1 has radian measure 1.

Since an angle of 360° corresponds to the circumference of a circle, or an arc of length 2π, we conclude that an angle with a degree measure of 360° has a radian measure of 2π.

 

The Six Basic Trigonometric Functions

To define the trigonometric functions, first consider the unit circle centered at the origin and a point P = (x, y) on the unit circle. Let θ be an angle with an initial side that lies along the positive x-axis and with a terminal side that is the line segment OP. We can then define the values of the six trigonometric functions for θ in terms of the coordinates x and y.

Let P = (x, y) be a point on the unit circle centered at the origin O. Let θ be an angle with an initial side along the positive x-axis and a terminal side given by the line segment OP. The trigonometric functions are then defined as

If x = 0, secθ and tanθ are undefined. If y = 0, then cotθ and cscθ are undefined.

We can see that for a point P = (x, y) on a circle of radius r with a corresponding angle θ, the coordinates x and y satisfy

Let θ be one of the acute angles. Let A be the length of the adjacent leg, O be the length of the opposite leg, and H be the length of the hypotenuse. A, H, and O satisfy the following relationships with θ:

 

Trigonometric Identities

A trigonometric identity is an equation involving trigonometric functions that is true for all angles θ for which the functions are defined. We can use the identities to help us solve or simplify equations.

1. Reciprocal identities

2. Pythagorean identities

3. Addition and subtraction formulas

4. Double-angle formulas

 

 

Graphs and Periods of the Trigonometric Functions

Let P = (x, y) be a point on the unit circle and let θ be the corresponding angle . Since the angle θ and θ +2π correspond to the same point P, the values of the trigonometric functions at θ and at θ +2π are the same. Consequently, the trigonometric functions are periodic functions. The period of a function f is defined to be the smallest positive value p such that f(x + p) = f(x) for all values x in the domain of f. The sine, cosine, secant, and cosecant functions have a period of 2π. Since the tangent and cotangent functions repeat on an interval of length π, their period is π.

 

Just as with algebraic functions, we can apply transformations to trigonometric functions.

The graph of y = cosx is the graph of y = sinx shifted to the left π/2 units. Similarly, we can view the graph of y = sinx as the graph of y = cosx shifted right π/2 units, and state that sinx = cos(x − π/2).