Warm-up! 1.State the domain and range of the relation {(-2,4), (1,3), (-2,-2)}. Is the relation a function? 2.Find [f○g](x) and [g○f](x) for f(x) = 2x – 4 and g(x) = x 2 3.Write the equation in slope-intercept form for the line with a slope of 6 and passing through the point (-3,4). 4.Graph f(x) = 4 – x 5.Graph 2x – y < 7

Welcome to Unit 8! Chapter 7: “Hat Trig!” In this unit we will learn… 8.1 simplify trigonometric expressions (7-1) 8.2 verify trigonometric equations (7-2) 8.3 use sum and difference identities to solve problems (7-3) 8.4 use half and double angle identities to solve problems (7-4) 8.5 solve trigonometric equations (7-5)

8.1: simplify trigonometric expressions (7-1) In this section we will answer… Just what is a trig identity? What are the basic trig identities? How can I use them to simplify expressions?

What is a trig identity? A trig identity is an equality between two trig expressions which is true for ALL values of the variable. To disprove an identity all you have to do is find one value that does not work in the equality.

What are the basic trig identities? Many of these you already know or can easily find!

Remember these?

The Quotient Identities

Simplify:

Pythagorean Identities x y r Let’s build them from scratch!

Are these the same?

Let us try to simplify!

Using Identities to find trig values Choose an identity which uses both tangent and secant. Substitute for tan x. Solve for secant. Determine the sign needed by the quadrant. Could also be found using a right triangle and Pythagorean theorem.

Using Identities to find trig values

Homework and Upcoming Events: p 427 #19 – 35 and 39 – 53 odd Reassessments due Wednesday!

Warm-up: p A38 Lesson 7-1 all

Homework:

8.2: verify trig equations (7-2) In this section we will answer… Are there more identities possible? How can I be sure that an identity is really true without trying every angle on earth? Can I use identities to find actual answers to problems?

Are there more identities possible? There are an any number of identities possible. As long as you build them based on existing identities you can make them all day!

How can I be sure that an identity is really true without trying every angle on earth? Algebraic Manipulation and Basic Trig Identities The “Do’s”:  DO simplify the more complicated side.  DO substitute (a lot!) using Basic Trig Id’s.  DO factor if possible.  DO multiply numerators and denominators by the same trig function.  DO find common denominators when you can. The “Don’t’s”:  DON’T work both sides of the equation at once!  If you want to switch, STOP working that side and move to the other.  DON’T move anything from one side to the other!

Let’s try one! Verify the following id.

Verify the following identity.

Can I use identities to find actual answers to problems? Sure can! Watch! Goal! Get one trig function = a number

Another… Find a numerical value of one trigonometric function of x.

A couple word problems… p 435 #40 and 42

Homework and Upcoming Events: p 434 #13 – 39 odd and 43 Quiz on tomorrow! Unit 8 Test after break.

Warm-up:

Homework:

Section 7-3: Sum and Difference Identities In this section we will answer… How can I manipulate an expression using trig identities so that I can solve it without a calculator?

Our handy-dandy handout!

Homework: p 442 #15 – 39 odd 43 Unit 8 Test after the break!

Warm-up: Flashcards!

Homework:

In this section we will answer… What other methods do I have to find trig ratios without a calculator? Section 7-4: Double Angle and Half Angle Identities

Question: Does ?

DOUBLE ANGLE Ids: If  represents the measure of an angle, then the following identities hold for all values of  :

ex. If, and the angle is in the first quadrant, find the exact value of the following:

If  represents the measure of an angle, then the following identities hold for all values of  : **Choose + or – based on ___________________________**

ex. Find an exact value for sin 195 .

ex. Verify:

Homework: p 454 #15-35 odd and 38 Test for Unit 8 – Tuesday FINAL: Open-Ended on Friday, June 13 th Multiple Choice on Tuesday, June 17 th

Warm-up: A38 Lesson 7-4 all Get your homework out!

Homework:

Section 6-8 and 7-5: Solving Trig Equations In these sections we will answer… How can I use trig ratios and trig identities to solve equations?

Principle Values Can generate a positive or a negative angle for a given trig ratio. Uses the first quad and an adjacent quad. Your calculator works in principle values. Designated by a capital letter:

Which Quads?

Start simple… Do odds in degrees/evens in radians

Let's step it up! Solve the equation for the principal values of x. Express your solutions in degrees.

Solve the equation for all values of θ from 0 ≤ θ < 2π

Solve the equation for the principal values of y in degrees.

Solve the equation for all real values of β in radians.

Solving by graphing

Partner Work: One piece of paper per partnership. Partners will work together to write the first problem. After that, one person solves, the other coaches/praises then switch. Only one person at a time is working. P459 #4 – 14 all

Homework: P459 #17 – 47 odd, 57 and 58 Test Wednesday!!! 7-1 to 7-5