Pre-Calculus B, 2nd Hour, Winter 2012: 2011

Wednesday, December 14, 2011

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.)

Monday, December 12, 2011

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

SINE

This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

COSINE This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

TANGENT This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

*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: This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

Proof of the COSINE formula: This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

Proof of the TANGENT formula: This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

*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 This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

)

6. x =

(eliminate duplicate solutions)

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

Sunday, December 11, 2011

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( This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

/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 This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

/12. We will use ( This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

/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 = This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

/3 and y = This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

/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 ( This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

/4) - (

/4).

7. The final answer is This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.