Chapter 5.5

• Power-Reducing Formulas

*The following power-reducing formulas can be derived from the double-angle formulas.

• Half-Angle Formulas

*You can derive the half-angle formulas from the power-reducing formulas by replacing x with x/2.

• Product-to-Sum Formulas

*each of the following product-to-sum formulas can be verified using the sum and difference formulas.

• Sum-to-Product Formulas

*Sometimes it is useful to reverse the procedure and write a sum of trigonometric functions as a product. This can be done with the following sum-to-product formulas.

Mr. Wilhelm-
I was unable to comment in the comment box, so I have posted my response:

I find simplifying and solving multiple-angle equations to be challenging. For example, knowing where to start and how to continue on when solving equations such as cos5x-7cos2x=8 is difficult as these problems may become very long, and many times lead to dead ends. In addition, several different identities must be substituted in for other terms in order to solve these problems. There are often several identities to choice from, and some identities may equal more than one thing, as does cos2x. Deciding which identity to substitute in is often difficult.

Daynna Myers

Chapter 5 Muddiest Points

What about Chapter 5 is most confusing for you?

(Answer this question by commenting on this post.)

Section 5.5 (part 1, pages 411-413): Double-Angle Formulas

SINE

COSINE

TANGENT

*These formulas are the most commonly used multiple-angle formulas. Like other trigonometric identities, they are used to rewrite trigonometric functions in more convenient forms.

Proof of the SINE formula:

Proof of the COSINE formula:

Proof of the TANGENT formula:

*Compared to others, the proofs for the double-angle formulas are simple. Their basis are the Sum & Difference Formulas for sine, cosine, and tangent (Section 5.4, page 404).

Practice Problem (page 411): Solving a Double-Angle Equation

Find all solutions of 2cosx + sin2x = 0

1. 2cosx + 2sinxcosx = 0 (substitute the SINE double-angle formula for sin2x)

2. 2cosx(1 + sinx) = 0 (factor out a 2cosx)

3. 2cosx = 0, 1 + sinx = 0 (set both factors equal to 0)

4. cosx = 0, sin = -1 (simplify both factors)

5. x = , x = (use prior knowledge of the unit circle to find all solutions for x on the interval )

6. x = (eliminate duplicate solutions)

*Depending on the period of the equation, there may be additional answers.

5.4 Sum and Difference Identities

Sum and difference identities are most commonly used when you have to find an obscure angle that isn't obvious on the unit circle.

Sum and Difference identities for sine, cosine, and tangent functions:

Proof of sin(x + y) = (sinx cosy) + (cosx siny)

Sin(x+y) = DE/AD

Sin(x+y) = (DF+FE)/AD

Sin(x+y) = (DF+BC)/AD

Now solve for the x's and y.

▲BDF: cosx = DF/DB

DF = DB(cosx)

▲ABC: sinx = BC/AB

BC = AB(sinx)

▲ABD: cosy = AB/AD

AB = AD(cosx)

Now substitute

(DF+BC)/AD = (DBcosx+ABsinx)/AD

(DF+BC)/AD = ((ADsiny)

The AD's all cancel out to leave you with

(sinx cosy) + (cosx siny)

Example problem:

Find the exact value of sin(/12)

Because /12 is kind of an obscure angle on the unit circle, you can use an identity to solve it instead of the unit circle.

1. Find two angles on the unit circle that you ARE familiar with and either add or subtract them to get /12. We will use (/3 - /4).

2. Now we know that sin(/12) = sin(/3 - /4).

3. Because this equation involves the subtraction of sine functions, we will use the difference identity for sine which is sin(x - y) = (sinxcosy) - (cosxsiny).

4. Since we know that x = /3 and y = /4, plug them into the the equation to get (sin(/3)cos(/4)) - (cos(/3)sin(/4)).

5. Simplify to get (/2)(/2) - (1/2)(/2).

6. Simplify again to get (/4) - (/4).

7. The final answer is

5.3 Solving Trigonometric Equations

How to Solve a Trigonometric Equation:


Extracting Square Roots Factoring Trigonometric Equation
1.
2.

However there are more difficult equations in which you must use the Quadratic Formula :
Functions of Multiple Angles
1. 2.