4.5 Graphs of Sine and Cosine Functions

Basic Sine and Cosine Waves

A sine wave is the graph of a sine function.  

y=sin x

Consider the table of values:

 The graph of y=sin x completes on cycle from [0,2π].  This makes the period of y=sin x 2π.

The domain of y=sin x is [-∞,∞].

The range is [-1,1].  This makes the amplitude of y=sin x 1.

A cosine wave is the graph of a cosine function. 

y=cos x

Consider the table of values:

The graph of y=cos x completes one cycle from [0,2π].  This makes the period of y=cos x 2π.

The domain of y=cos x is [-∞,∞]. 

The range is [-1,1].  This makes the amplitude of y=cos x 1.  

Transformations of Sine and Cosine Functions 

Consider the functions y = a sin[b(x-c)] + d and y = a cos[b(x-c)] + d where a, b, c, and d are all constants.  These constants affect the graphs of the parent functions y=sin x and y=cos x in the following ways:

a = vertical stretch factor

b = horizontal stretch factor

c = horizontal shift (phase shift)

d = vertical shift (midline shift)

Amplitude and Period of Sine and Cosine Waves

The amplitude of y=sin x and y=cos x represents half the distance between the maximum and minimum values of a function and is given by

Amplitude = |a| 

The period of y=sin x and y=cos x is the time it takes to complete one cycle of the unit circle and is given by

Period = 2π / |b|

Even and Odd Functions

Sine is an odd function:

A function f is odd if, for each x in the domain of f,

f(-x) = -f(x)

To prove this,

sin(-π/2) = -sin(π/2)

          -1 = -(1)

          -1 = -1

Also, the graph of y=sin x is symmetric about the origin.  

Cosine is an even function.

A function f is even if, for each x in the domain of f,

f(-x) = f(x)

To prove this,

cos(-π) = cos(π)

        -1 = -1

Also, the graph of y=cos x is symmetric about the y-axis.  

4.3 Right Triangle Trigonometry

Trigonometric Functions

In any right triangle, all three sides have a relation with any of the 3 angles. Relative to angle x, these three angles are either adjacent to angle x, opposite of angle x, or the hypotenuse of the triangle.

Using this principle, 6 different ratios can be created. It is also important to note that angle x is greater than 0 degrees and less than 90 degrees.

SOH-CAH-TOA

                      

                      

The functions cosecant, secant, and cotangent are reciprocals of sine, cosine, and tangent respectively. These are known as Reciprocal Identities.

                             

The cotangent function and tangent are known as Quotient Identities

       

This information, along with the Pythagorean Theorem can be used to to find the Pythagorean Identities:

          

Lets derive the first identity:  by using the Pythagorean theorem and our knowledge of the trigonometric functions. 

           

           

Trigonometry and the Unit Circle

Looking at the first quadrant of the unit circle, imagine that a right triangle is created with an angle measure of  at the origin. Using the property triangles, it can be inferred that the remaining angle is . By using the trigonometric functions, either leg of the triangle can be determined. To find the leg opposite of angle x, you can use the sine function to determine the side length. 

     Because we know that angle x is , we replace x with the degree of the angle. Since the the radius of the unit is 1 unit, we can say that the hypotenuse of the triangle is 1. 

      Here we can see that the sine of  is equal to the side opposite of the angle. This  value also corresponds to the y value of the point that lies on the unit circle. Therefore,  is equal to the y value of the point created where the hypotenuse, and the edge of the circle intersect. 

Using the same logic, it can be seen the the  is equal to the side adjacent to angle x. Therefore,  is equal to the x value of the point created where the hypotenuse and the edge of the circle intersect.

This information can be used to find any point on the unit circle if the corresponding angle is known.

                                         

4.1 Radian and Degree Measure

What is an angle?

An angle is simply two rays that share a common endpoint. An angle is a vector meaning it has magnitude and direction. The starting position of a ray is refereed to as the initial side of the angle and the position remaining after rotation is refereed to as the terminal side. The common end point where the rays meet is the vertex.

Envisioning a coordinate system, an angle is in standard position if the initial side of the angle lies on the positive x-axis.

Positive angles result from counterclockwise rotation and negative angles result from clockwise rotation.








Angles are often labels with upper case or Greek letters. Angles are coterminal if they share a common initial and terminal side.

In this case, for all three angles created by the blue, red, and green lines, the resulting angles are coterminal.

What are we measuring when we measure an angle?
To measure an angle means to measure the amount of revolution. 

Since the circumference of a circle is 2πr, a central angle containing one full revolution counterclockwise creates an arc length s, equal to 2πr. This means that 2π radians correspond to 360° and π radians correspond to 180° and so on. 

If one circle (360°) is also 2π radians, there must be about 6.26 radians in one full circle. 

A circle can be divided and measured the following ways with the Unit Circle:

What are radians?
Radians are another way to measure angles besides degrees. Envision a central angle θ on a circle centered around the origin showing a vertex at the origin. A single radian is the amount of rotation needed to make the length of the intercepted arc s equal to the radius r.
If s = r then, θ =1 radian.









To determine arc length a proportion can be used.

Where...
Angle Measure: θ
Total Measure of One Circle: 360 (in degrees)  or 2π (in radians)
Arc Length: s (if degrees or radians are used for the total measure of one circle that same unit will remain for arc length)
Circumference: 2πr

Similarly, to convert from radians to degrees and vise versa a proportion is used.

Multiply the magnitude by... 

to go from degrees to radians and... 

to go from radians to degrees. The original unit will cancel out and the desired unit will remain.