4.5 Graphs of Sine and Cosine Functions

The amplitude of y = a sin x and y = a cos x represents half the distance between the maximum and minimum values of the function (the distance from the maximum to the midline or the distance from the minimum to the midline).
The amplitude = 
The period of y = a sin bx and y = a cos bx, given b is a positive real number, is equal to 


The Graph of Sine

The period is 2π and the range is [-1,1].
As shown in the graph, the sine function is odd.
y = a sin (bx - c) + d
amplitude = 

period  = 

phase shift (shift left/right) = c

midline shift (shift up/down) = d

The Graph of Cosine

The period is 2π and the range is [-1,1].
As shown in the graph, the cosine function is even.
y = a cos (bx - c) + d
amplitude = 

period  = 

phase shift (shift left/right) = c

midline shift (shift up/down) = d

4.3 Right Triangle Trigiometry

Section three of chapter four has to do with trigonometric identities, and how they can be manipulated to create other identities.

An identity is defined as being an equation that for every possible input for the variables, the equation is true. In dealing with trigonometric functions, the variable is quite often theta.

Some basic trig. identities are the reciprocal identities, which express the relationship of sin, cos, and tan, and their respective inverses.

Quotient identities have to do with the relationship between cot and tan, with sin and cos.
 

The even/odd identities show whether each trig. function is even or odd.

Even/Odd Identities



Note: cos and sec are the only odd functions; the rest are even.

The next group of identities is referred to as the 'Pythagorean Identities', and can only be truly understood when understanding how they are derived from a right triangle.

 

 

(Pythagorean theorem)


<------Most important Pythagorean identity

From the equation above, the other two Pythagorean identities can be derived.

Pythagorean Identities









Application

When solving problems using the identities listed above, one if often asked to manipulate one side of the equation. It is probably in one's best interest to work on the more complicated side to get the simpler identity on the other side.

There is no one, correct way to do these problems, so one can be creative with their work.

Examples:











 

4.2 Trigonometric Functions: The Unit Circle

The Unit Circle is a circle with a radius of one unit about the origin.









(https://www.wolframalpha.com/input/?i=x%5E2%2By%5E2%3D1)






The angle measures of the Unit Circle are measured in degrees and radians. For our purposes, we will stick to radians.

One revolution of the unit circle is  radians.

On the Unit Circle, it is important to know what point value is located at certain radians.

The radians to know are
 and .

While each radian is associated with a unique point, there are only three points to know in order to know all of the values for the aforementioned radians, and those are the points in the x,y plane. This is because the rays that form the angles in radians of the x,y plane are reflections across axes. When you want to know the points in the -x,y plane, simply take the point of the complementary angle in the x,y plane and make the x-value a negative. When dealing with the -x,-y plane, make both values of the complementary angle's points negative. Finally, when working with the x,-y plane, make the y values of the complementary angles negative.





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With  and the Unit Circle, we can determine the values of the Trigonometric Functions.

The Trig Functions are as follows:

Sine (sin)             Cosecant (csc)
Cosine (cos)         Secant (sec)
Tangent (tan)       Cotangent (cot)

Each Trig Function has a unique relationship with the points and angles of the Unit Circle. Let  be a real number and let (x,y) be a point on the Unit Circle corresponding to .

                           

                      

                     

As you may notice, the Trig Functions also have a relationship with each other. The Functions on the left and the functions on the right are reciprocals. The reciprocal relationships are between sin and csc, cos and sec, and tan and cot.

Note: When , tan and sec are undefined. When , cot and csc are undefined.

Remember that taking the trig function when  IS possible for all functions except for csc and cot.

The domain of both sin and cos is .

The range of both sin and cos is . This is because the radius of the Unit Circle is 1.

Adding  to each value of  in the interval  completes a second revolution around the Unit Circle, thus  and  correspond to  and . The same is true for repeated revolutions, whether they are positive or negative, which means:





for any integer n and real number . This behavior is known as periodic.

The Trig Functions, like any function, are defined as even or odd.

Even:

             

Odd:

          

         

ON A CALCULATOR be sure to set the mode from degrees to radians. ALSO there are no buttons for csc, sec, or cot. However, since these are the reciprocals of sin, cos, and tan, you can get by simply by dividing one by whichever function you need to produce the reciprocal, or by raising the function of the reciprocal to the negative first power.

Example:

         or         



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