5.5 Double-Angle Formulas

The double-angle formulas are some of the most commonly used and most important trigonometric identities. These are the formulas and how to derive them:

To derive the sine formula, you can use a sum formula:

To derive the cosine formulas, follow a similar process: 

The above formula is also equivalent to this: 

Simplifying, you get

You can also change the first formula to this:

Again, simplify:

To derive the tangent formula, use a sum formula:

Examples:

Solving a Double-Angle Equation

Find all solutions to the following equation on the interval 

1) Begin by using a double-angle formula for cosine:

2) Next, factor.

3) Set the factors equal to zero and simplify.

    

4) Solve for x on the given interval.

    

Using Double-Angle Formulas in Sketching Graphs

Graph the following function over the interval  

1) First, factor out a 2.

2) Then, you can substitute using a double-angle formula.

From this equation, we know that the amplitude of the graph is 2, and the period is .

Using this information, we can sketch the graph: 

Evaluating Functions Involving Double Angles

Use the following to find .

     

1) From the given information, you can draw a triangle like this:

2) Next, you use the double-angle formula, substitute in the known values, and simplify.

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.