Properties of Trig Functions


About Trigonometric Functions: Trigonometric functions occur in any real world situation that continuously recurs after a certain point in time. Such real world examples could be the rise and fall between high and low tide, radio waves, electrical currents, the orbit of the earth, or even the spinning of a ferris wheel. These real world examples of sine and cosine functions are represented in a continuous graph that looks like waves. This occurs because of the recurrence of the functions periods. A period is the distance required for the function to complete one full cycle. For example, the time it takes for the highest point the tide reaches all the way to the lowest point of the tide and back to the highest point is one period. Another example is the time it takes for a ferris wheel to go around one time. Other properties are used as well to describe trigonometric functions such as the vertical shift of the function, the amplitude of the function, and phase shift of the function described below.

The graph of a trigonometric function is represented by:

Y= a +(-) b SIN/COS(cX +(-) d)

a=Vertical Shift
b=Amplitude
c=2π/period
d=(phase shift)/c

+b = no reflection
-b = reflection
+d= horizontal shift left
-d= horizontal shift right


Vertical Shift

The vertical shift moves the graph vertically, up or down. This movement is based on what happens to the y-value of the graph. The value of the vertical shift is the value of the graphs midline (center).

vertical-shift.gif
midline.jpg

Amplitude

The amplitude vertically stretches the function. the value of the amplitude in the function is the units from maximum point to the midline or minimum point to the midline.

amplitude-of-the-wave.png

Period

The period is the distance required for the function to complete one full cycle.

illustration-graph-of-period-of-sine-of-x-labelled-v2.png

Phase Shift

Phase shift is the amount of horizontal displacement of the function from its original position.

phaseabc.jpg

Reflection

The reflection of a trigonometric graph flips the graph over the x axis. This changes all the y values of the function. The reflection is represented by a negative on the amplitude in the trigonometric equation.

graph_of_cos_reflection.png

Transformations on Trig Functions

Trig-Transformation-Formulas.jpg

How to Graph Trig Functions

When Graphing Trig Functions like Sine there are 5 key point to know, so you have the curve right. The Starting Point (SP), the End Point (EP), the Middle Point (M), the Quarter 1 Point (Q1), and the Quarter Three Point (Q3). With these five points, and the knowledge of how a sinusoidal functions should look, you can graph a sinusoidal function. (Period is always represented by P)

(1)
\begin{equation} SP = Phase shift \end{equation}
(2)
\begin{equation} EP = SP+P \end{equation}
(3)
\begin{equation} M = (SP+EP)/2 \end{equation}
(4)
\begin{equation} Q1 = (SP+M)/2 \end{equation}
(5)
\begin{equation} Q3 = (M+EP)/2 \end{equation}
Key Trig Values for Graphing Without Transformations
Trig-T-charts.png

Types of Trig Functions


Sine

graph_sin_pi.gif

As Degrees

Angle Sine of Angle Angle Sine of Angle
0 180° 0
30° 1/2 210° -1/2
45° √2/2 225° -√2/2
60° √3/2 240° 1√3/2
90° 1 270° -1
120° √3/2 300° -√3/2
135° √2/2 315° -√2/2
150° 1/2 330° -1/2

As Radians

Angle Sine of Angle Angle Sine of Angle
0 0 π 0
π/6 1/2 7π/6 -1/2
π/4 √2/2 5π/4 -√2/2
π/3 √3/2 4π/3 1√3/2
π/2 1 3π/2 -1
2π/3 √3/2 5π/3 -√3/2
3π/4 √2/2 7π/4 -√2/2
5π/6 1/2 -1/2

Cosecant

graph_csc_pi.gif

Cosine

graph_cos_pi.gif

Secant

graph_sec_num.gif

Tangent

graph_tan_pi.gif

Cotangent

graph_cot_pi.gif

Practice Problems

A. writing_the_equation_for_a_given_trig_graph_clip_image002.jpg
Write an equation for the graph in the form y = A sin ( Bx + C ).

Process

The amplitude of the graph is 2, so A = 2.

We know that one typical complete wave of the sine function starts at a y-value of 0, increases to 1, decreases through 0 and on to –1, and then increases to a y-value of 0, so the interval [-π/3, 3π +2π/3] contains one wave length of the function. The length of the interval tells us that the period is (3π +2π/3) - (-π/3) = 4π.

Using the formula: period = 2π/b , we set 4π=2π/b = and solve for B , finding that b=1/2.
We have already observed from the graph that the phase shift is -π/3, so we can use the formula:
phase shift = -c/b, set -π/3=c/(½) and solve for C , finding that c=π/6.

The equation of the graph is: $y=2sin((½)x+π/6)$


B. writing_the_equation_for_a_given_trig_graph_clip_image002.jpg
Write an equation for the graph in the form y = A cos ( Bx + C ).

Process

The amplitude of the graph is still 2, so A = 2.

We know that one typical complete wave of the cosine function starts at a y-value of 1, decreases through 0 to –1, and then increases through 0 to 1, so the interval [2 p /3, 4 p +2 p /3] contains one wave length of the function.

The length of the interval tells us that the period is 4 p +2 p /3 - 2 p /3 = 4 p , the same period as we found in Example 1, so b=½ as before.
We can use a phase shift of 2π/3, so the formula: phase shift = -c/b becomes 2π/3=-c/(½) and c= -π/3.

The equation of the graph is also: $y=2cos((½)x-π/3)$


c.

Graph two periods of: g(x)=sin(πx+​2​​π​​)+3

Process

The amplitude of this graph is going to be the same as for regular sine waves, because there's 1 multiplied on the sine.
The midline of the graph is going to be at y = 3 instead of y = 0 because of the "+3" at the end of the function.
The regular period for sine waves is 2π, but the variable in this function is multiplied by π; doing the division, the period of this particular function is going to be 2​​π​​/π​​=2.
Now the phase shift. The phase shift = d/c. (π/2)/π=½
graphs27.gif