5.5- Double Angle Formulas.

We learned 3 new identities today. They're pretty helpful, and we should definitely know them.

These formulas can be used when the variable angle, X, is doubled as shown below.

We can derive these formulas using the some identities we learned last section.

sin(2x) = sin(x+x)

= sin(x) cos(x) + cos(x) sin(x)

= 2 sin(x) cos(x)

cos(2x)= cos(x+x)

= cos(x) cos(x) - sin(x) sin(x)

= cos^2 (x) - sin^2 (x)

which can also be written as:
= 2 cos^2 (x) -1

= 1- 2 sin^2 (x)

tan(2x)= tan (x+x)

= (tan (x) + tan (x)) / (1-tan(x) tan(x))

=(2 tan(x)) / (1- tan^2 (x))

Therefore, these identities can be summed up as the following:

1- sin(2x) = 2 sin(x) cos(x)

2- cos(2x) = cos^2 (x) - sin^2 (x)

3- tan(2x) = (2 tan(x)) / (1- tan^2 (x))

There you go. Use these identities to do what you please, and have a ton of fun.

5.4: Sum Difference Formulas

In this section, we study the use of the six Sum and Difference Formulas. Below are the 6 formulas.

sin(A+B) = sin A cos B + cos A sin B

sin(A−B) = sin A cos B − cos A sin B

cos(A+B) = cos A cos B − sin A sin B

cos(A−B) = cos A cos B + sin A sin B

tan(A+B) = (tan A + tan B) / (1 − tan A tan B)

tan(A−B) = (tan A − tan B) / (1 + tan A tan B)

This formulas all can be used to evaluate trigonometric functions, proving identities, solving trigonometric equations, solving a sum formula, and deriving reduction formulas. They all should be memorized and can easily be done so by memorizing them in groups based on sine, cosine, and tangent.

If you are interested in how these formulas were derived, you can check out the following page by clicking here showing the derivation of the Sum and Difference Formulas.

Solving Trig Equations

When solving trigonometric equations, there are two things to consider:

1) You may need to square both sides in order to solve it.

2) Checking for extraneous solutions is necessary in some cases.

Squaring Both Sides

It may be necessary to square both sides of the equation in order to solve it. For example, if left with something like 1+ cosx=sinx, there are no other options.

1 + cosx = sinx

1 + cos^2x = sin^2x *Square both sides.

1 + cos^2x= 1 - cos^2x *Substitute the pythagorean identity.

cos^2x = -cos^2x *Subtract 1 from both sides.

2cos^2x= 0 *Add cos^2x to both sides.

cosx = 0 * Divide by 2 and take the square root of both sides.

x = π/2, 3π/2 *Use unit circle knowledge to find angles

and now....

We must check for extraneous solutions!

When is this required?

1) Anytime you square both sides of the equation.

2) To make sure the denominator is not 0.

For #1, plug in you're answers to make sure it remains true.

For #2, plug in answers to the denominator, and make sure it just does not equal 0.

5.2 & 5.3

In these two sections we are solving equations for a variable, such as x. This is very similar to what we have done in previous years and sections. Except, we are adding the trigonomic ideas that we have been focusing on in the previous sections.

SIMILARITIES

We are still focusing on getting x by itself.
This means that the order of operations is the same- addition/subtraction first, then multiplucation/division, and finally parenthesis/exponents.

ex. of old equation ex. of new equation
2x-5=5 2cos(x)-5=5

DIFFERENCES

Now we have to use the ideas of sine, cosine, tangent, secant, cosecant, and cotangent.
The answers we get from these equations are angle measures from the unit circle.
Instead of getting one answer from a simple equation, we usually get multiple answers.

ex.
2cos(x)+1=0
2cos(x)=-1
cos(x)=-1/2
x=arccos(-1/2)
x=4π /3 x=2π /3

These new ideas use the concepts that we have already learned and incorporated them together.

Fun with Trig Identities

Welcome to 2nd Tri Pre Calc! So the focus right now is on Identities, and when dealing with identities there are a couple of FUNdamental identities you need to know. They are as follows:

Reciprocal Identities

Quotient Identities

Pythagorean Identities

Cofunction Identities

Even/Odd Functions

Know all of these and you are golden.