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Day 55 Verifying Identities 5.1 & 5.2

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Day 55Verifying Identities5.1 & 5.2

- Recognize and write the fundamental trigonometric identities.
- Use the fundamental trigonometric identities to evaluate trigonometric functions, simplify trigonometric expressions, and rewrite trigonometric expressions.

- What are we doing in this chapter?
- Review of identities from Chapter 4
- Techniques for verifying identities
- Homework

Solve for x:

1. x + 3x – 4 = 2x – 7

2. (x + 1)2 – 3 = 4x + 1

Find the zeros of the function:

3. y = 3x +7

4. y = x2 + 7x + 12

What did you do to solve the problem?

Our ultimate goal is to solve trigonometric functions.

To do so we have to be able to …

- Simplify
- Combine “like” terms (may need interpretation)
- Factor

We will also be verifying that expressions are equivalent or proving that a statement is an identity

What is an identity?

To be able to solve many trigonometric problems you must first simplify the expression.

Verifying identities means to demonstrate that two expressions represent the same thing. This allows you to replace one expression with another to help in simplifying.

Verifying and simplifying uses the same set of skills and techniques.

Simplifying: How simple is simple enough?

- No denominators
- All like terms combined
- All common factors have been eliminated

Here are some equivalent terms; which is more simple A or B?

- 5B. 2 + 3
A. 6/2B. 3

- 5/100B. 1/20
A. 20B. 4 • 5

Here are some equivalent terms; which is more simple A or B?

- 1B. sin2θ + cos2θ
A. sin θ/cosθ B. tan θ

- 1/sec θB. cosθ
A. 1+ tan2 θB. sec2 θ

What are some of the identities we have studied so far…?

- Reciprocal
- Co Function
- Quotient
- Pythagorean
- Even / Odd

sin = 1/csc csc = 1/sin

cos = 1/sec sec = 1/cos

tan = 1/cotcot = 1/tan

sin = cos(90 ) cos = sin(90 )

tan = cot(90 ) cot = tan(90 )

sec = csc(90 ) csc = sec(90 )

Quotient and Pythagorean Identities

Quotient Identities

tan = sin /cos cot = cos /sin

Pythagorean Identities

sin2 + cos2 = 1 tan2 + 1 = sec2 cot2 + 1 = csc2

- Cosine and secant functions are even
cos (-t) = cos tsec (-t) = sec t

- Sine, cosecant, tangent and cotangent are odd
sin (-t) = - sin tcsc (-t) = - csc t

tan (-t) = - tan tcot (-t) = - cot t

Adjustments to known identities:

sin2 + cos2 = 1

can also be written as:

sin2 = 1 – cos2 OR

cos2 = 1 – sin2

The other Pythagorean Identities can be similarly adjusted.

Checking with a calculator

There are several techniques or methods of approaching these problems based upon the structure of the problem…

- Substituting identities to eliminate like terms and simplify
- Splitting rational functions so it is easier to see what identities can be used to simplify
- Factoring
- For rational functions, finding a common denominator to help simplify
- Eliminating the denominator (creating binomials that can be simplified into a single term)
- Combinations of all the above

- These problems take practice to get good at them! Even if you are stumped, try something! Even a path that leads to a dead end can provide valuable insight.

- sec x cos x
- tan2 x – sec2x
- sin (-x) /cos (-x)

- cot (x) / csc (x)
- sec θ • sin θ / tan θ
- (1 + sin θ) / cos θ

Before moving on to factoring expressions that have trigonometric functions, let’s review factoring linear and quadratic functions.

Simple factoring: take out common factor

1. x2 - xy

2. 2x – 4xy

Factoring quadratics

3. x2 - 2x + 1

4. x2 - 3

- Simpletan2 x – tan2 x sin2 x
Now that it is factored, can it be simplified using identities

2. Trinomial tan4 x + 2 tan2 x + 1

Now that it is factored, can it be simplified using identities

2 sec2 x – 2 sec2 x sin2 x - sin2 x – cos2 x

- Simplifying is taking an expression that must be worked until it is in the simplest form:
- No denominators
- Combining like terms
- Eliminating common factors

- Verifying is taking an equation and make one side of the equation look like the other. For our activities we will work only one side of the equation. Never divide by a variable.

- 5.1 Page 359 15-26 all (matching – helps you see where you are going.)
- 5.2 page 367 1-13 odd, 27 – 30 all
(work one side of the equation only.)