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.