Transcript Section 1.5 Trigonometric Functions

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
y
1
-1
0
-1
1
x
The Unit Circle
1
1 radian
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
y
1
(cost, sint)
t
t radians
-1
0
1
1
x
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
-1
(cost, sint)
Assume the angles are ALWAYS in radians,
unless specified otherwise!
• So we define the following:
cost  x and sin t  y
• And have the resulting identity (Pythagorean
Identity)
cos2 t  sin 2 t  1
• 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
 1  cost  1 and  1  sin t  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
f (t )  A sin(Bt) and g (t )  A cos(Bt)
• 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
y  2 sin( 2x )  1
y  cos(2x  1)
Tangent Function
sin t
tan t 
cos t
• 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 sin( )   43 and cos(θ) > 0 find the
other 5 trig functions
• Find all the solutions for
2cos(2  1)  1  0
for 0 ≤ θ ≤ 2π