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.

5.3-Solving Trigonometric Equations

The goal of solving trigonometric equations is to isolate the trigonometric function involved in the equation, this can be done using standard algebraic techniques.

Solving a basic trig equation

2sin(x)-1=0                        Original equation

2sin(x)=1                        Add one to each side

sin(x)=1/2                        Divide each side by 2

Use the unit circle or sine wave to find the solutions to this equation

According to the unit circle and sine wave

Therefore  and  are the two solutions in the interval ⊏0,2∏).

 If you want to express all possible solutions for this trig equation add 2n∏ (because the period is 2∏)  to the end of each solution. For this particular equation, there are infinitely many solutions.  (n is an integer)

Factoring Trig Equations

By taking out the cot(x), you can then use the zero product property.

The solutions x=∏/2 and x=(3∏)/2 come from the equation cot(x)=0

This equation needs to be simplified further

 Add 2 to both sides and then take the square root of both sides.

*Watch out for equations that can have no solution because their solution is outside the range of the function.

The real solutions to this equation is: 

*Remember, the period of cot is ∏, not 2∏. If you want to express all possible solutions for this trig equation add n∏ to the end of each solution. For this particular equation, there are infinitely many solutions.  (n is an integer)

*It would be redundant for this to be the solution:(The first equation is the simplified solution it is the only solution necessary for the final answer.)

 Sum and Difference of Cubes

-Sometimes you will need to use the sum or difference formula in order to solve trig equations. It is helpful to remember the phrase: Same, opposite, always positive (S.O.A.P) when factoring.

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5.4- Sum and Difference Formulas

 

Consider  and .

Using sum and difference formulas: In this section, we study the use of several new trigonometric identities and formulas.  
Sum and difference formulas can be used to find the exact values of trigonometric functions involving sums or differences of special angles (angles whose sines, cosines, and tangents can be found on the unit circle).









Deriving the sum and difference formulas:

AND

AND

Substituting those values in, we find that...

we are able to divide out AD and conclude that

THIS is our first sum and difference formula. 

Assuming they are derived through a similar process, we must also know these formulas:

Using these formulas and our knowledge of the quotient identities, we can also derive the sum and difference formulas for the tangent function.  We know that:

using the formulas we just derived, we can say that:

multiplying both the top and the bottom by , and then simplifying, yields this formula:

To get the difference formula for the tangent function, you simply use

and use the same process, to yield a similar equation, but with different signs.

Example:

Find the exact value of the sine, cosine and tangent of an angle with the measure:

EVALUATING SINE

EVALUATING COSINE

EVALUATING TANGENT

Sum and difference formulas can also be used to prove trigonometric identities, as in this example:

Prove the confuction identity:

Using the formula for cos(x-y), you have

You can also use these identities to solve trigonometric equations, as in this example:

Find all solutions of 

on the interval

Using sum and difference formulas, rewrite the equation as