Section 1.5 Trigonometric Functions

The Unit Circle Circle with radius of 1 What is the circumference? 2π Therefore we say 360° = 2π radians So 1 radian = 180°/ π 1 -1 x y

The Unit Circle 1 radian Arc Length = 1 1 An angle of 1 radian is defined to be the angle at the center of the unit circle, which cuts off an arc of length 1 radius length, measured counterclockwise.

The Unit Circle (cost, sint) y 1 -1 x t radians t (cost, sint) The angle generated by rotating counterclockwise from the x-axis to the point P(t) on the unit circle is said to have radian measure t The coordinates of P(t) are (cost, sint) Assume the angles are ALWAYS in radians, unless specified otherwise!

So we define the following: And have the resulting identity (Pythagorean Identity) These are periodic functions A function is periodic if there exists a positive number T such that f(t) = f(t + T) For sine and cosine, the period is 2π As we travel around the circle, they oscillate between -1 and 1

Graphs of Sine and Cosine Notice they are periodic, only shifted π/2

Half the distance between the maximum and minimum values is called the Amplitude The Period is the smallest time needed for a function to execute one complete cycle In the last example we saw that multiplying sin(x) by 2 increased the amplitude Now let’s see how we can change the period We will plot sin(2x) Let’s see how the period changes

Multiplying x by 2 changed the period to π In general we have Where |A| is the amplitude and 2π /|B| is the period We can also shift the graphs vertically and horizontally Let’s look at a few examples

Tangent Function The tangent function gives the slope of the line through the origin and point P on the unit circle Let’s look at the graph Notice that is only has a period of π Notice that it has vertical asymptotes

Inverse Trig Functions We can tell by looking at the graphs that trig functions are not 1-1 But we can restrict our domain in order to make them invertible For -1 ≤ y ≤ 1, arcsiny = x means that sinx = y with –π/2 ≤ x ≤ π/2 For -1 ≤ y ≤ 1, arccosy = x means that cosx = y with 0 ≤ x ≤ π For any y, arctany = x means that tanx = y with –π/2 ≤ x ≤ π/2 arcsiny = sin-1y, arccosy = cos-1y, arctany = tan-1y

Examples Given and cos(θ) > 0 find the other 5 trig functions Find all the solutions for for 0 ≤ θ ≤ 2π