day 55 verifying identities 5 1 5 2
Download
Skip this Video
Download Presentation
Day 55 Verifying Identities 5.1 & 5.2

Loading in 2 Seconds...

play fullscreen
1 / 32

Day 55 Verifying Identities 5.1 & 5.2 - PowerPoint PPT Presentation


  • 135 Views
  • Uploaded on

Day 55 Verifying Identities 5.1 & 5.2. What you will learn…. Recognize and write the fundamental trigonometric identities. Use the fundamental trigonometric identities to evaluate trigonometric functions, simplify trigonometric expressions, and rewrite trigonometric expressions.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Day 55 Verifying Identities 5.1 & 5.2 ' - misha


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
what you will learn
What you will learn…
  • Recognize and write the fundamental trigonometric identities.
  • Use the fundamental trigonometric identities to evaluate trigonometric functions, simplify trigonometric expressions, and rewrite trigonometric expressions.
plan for the day
Plan for the Day
  • What are we doing in this chapter?
  • Review of identities from Chapter 4
  • Techniques for verifying identities
  • Homework
solving non trigonometric equations
Solving Non Trigonometric Equations

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?

solving trigonometric functions
Solving Trigonometric Functions

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
another piece
Another Piece

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

What is an identity?

simplifying equations and verifying identities
Simplifying Equations and Verifying Identities

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 equations and verifying identities1
Simplifying Equations and Verifying Identities

Simplifying: How simple is simple enough?

  • No denominators
  • All like terms combined
  • All common factors have been eliminated
which is more simple
Which is more simple?

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

  • 5 B. 2 + 3

A. 6/2 B. 3

  • 5/100 B. 1/20

A. 20 B. 4 • 5

which is more simple1
Which is more simple?

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

  • 1 B. sin2θ + cos2θ

A. sin θ/cosθ B. tan θ

  • 1/sec θ B. cosθ

A. 1+ tan2 θ B. sec2 θ

what are identities
What are Identities?

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

  • Reciprocal
  • Co Function
  • Quotient
  • Pythagorean
  • Even / Odd
reciprocal functions
Reciprocal Functions

sin  = 1/csc csc = 1/sin

cos = 1/sec sec = 1/cos

tan = 1/cot cot = 1/tan

cofunctions
Cofunctions

sin  = cos(90  ) cos  = sin(90  )

tan  = cot(90  ) cot  = tan(90  )

sec  = csc(90  ) csc  = sec(90  )

slide14

Quotient and Pythagorean Identities

Quotient Identities

tan  = sin  /cos  cot  = cos  /sin 

Pythagorean Identities

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

even and odd trig functions
Even and Odd Trig Functions
  • Cosine and secant functions are even

cos (-t) = cos t sec (-t) = sec t

  • Sine, cosecant, tangent and cotangent are odd

sin (-t) = - sin t csc (-t) = - csc t

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

using identities we know
Using Identities we know

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.

verifying identities using technology
Verifying Identities using Technology

Checking with a calculator

simplifying or verifying algebraically how
Simplifying or Verifying Algebraically: How?

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

simplifying or verifying algebraically how1
Simplifying or Verifying Algebraically: How?
  • 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
keep in mind
Keep in Mind!
  • 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.
using the identities to simplify
Using the Identities to Simplify
  • sec x cos x
  • tan2 x – sec2x
  • sin (-x) /cos (-x)
splitting rational functions then use the identities
Splitting Rational Functions …then use the identities
  • cot (x) / csc (x)
  • sec θ • sin θ / tan θ
  • (1 + sin θ) / cos θ
factoring
Factoring

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

factor
Factor

Simple factoring: take out common factor

1. x2 - xy

2. 2x – 4xy

Factoring quadratics

3. x2 - 2x + 1

4. x2 - 3

factoring with trig functions
Factoring with Trig Functions
  • 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

combination multiple manipulations
Combination Multiple Manipulations

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

simplifying vs verifying
Simplifying vs Verifying
  • 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.
homework 29
Homework 29
  • 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.)

ad