1. Section 1.5 Trigonometric Functions

2. 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

3. 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.

4. 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!

5. 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

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

8. 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

10. 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

14. 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

16. 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

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