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1 Algebra2/Trig Chapter 13 Packet In this unit, students will be able to: Use the reciprocal trig identities to express any trig function in terms of sine, cosine, or both. Prove trigonometric identities algebraically using a variety of techniques Learn and apply the cofunction property Solve a linear trigonometric function using arcfunctions Solve a quadratic trigonometric function by factoring Solve a quadratic trigonometric function by using the quadratic formula Solve a quadratic trigonometric function containing two functions by using identities to replace one of the functions. Name:______________________________ Teacher:____________________________ Pd: _______

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### Transcript of Algebra2/Trig Chapter 13 Packet - White Plains Public ... Chapter 13: Sections 1 - Solving First...

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Algebra2/Trig Chapter 13 Packet

In this unit, students will be able to:

Use the reciprocal trig identities to express any trig function in terms of sine, cosine, or both.

Prove trigonometric identities algebraically using a variety of techniques

Learn and apply the cofunction property

Solve a linear trigonometric function using arcfunctions

Solve a quadratic trigonometric function by factoring

Solve a quadratic trigonometric function containing two functions by using identities to replace one of the functions.

Name:______________________________

Teacher:____________________________

Pd: _______

2

Day 1: Solving First Degree Trig Equations

SWBAT: Solve First Degree Trig Equations

Pgs. 3 – 7 in Packet

HW: Pgs. 8 – 10 in Packet

Day 2: Trig Equations by Factoring

SWBAT: Solve Trig Equations by Factoring

Pgs. 11 – 14 in Packet

HW: Pgs. 15 – 17 in Packet

Day 3: Unfactorable Trig Equations

SWBAT: Solve Second Degree Trig Equations using the Quadratic Formula

Pgs. 18 – 21 in Packet

HW: Pgs. 22 – 23 in Packet

***Quiz after Day 3***

Day 4: Solving Trig Equations With More Than One Function

SWBAT: Solve trigonometric equations using reciprocal identities

Pgs. 24 – 28 in Packet

HW: Pgs. 29 – 31 in Packet

Day 5: Solving Trig Equations With More Than One Function

SWBAT: Solve trigonometric equations using Pythagorean’s identities

Pgs. 32 – 35 in Packet

HW: Pgs. 36 – 38 in Packet

Day 6: Solving Trig Equations With More Than One Function

SWBAT: Solve trigonometric equations using Double-Angle identities

Pgs. 39 – 46 in Packet

HW: Pgs. 47 – 51 in Packet

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Chapter 13: Sections 1 - Solving First Degree Trigonometric Equations

SWBAT: Solve first degree trig equations

Warm - Up:

Identify trig values of quadrantal angles

Sine

Cosine

Tangent

Sin 0/360 =

Sin 90 =

Sin 180 =

Sin 270 =

cos 0/360 =

cos 90 =

cos 180 =

cos 270 =

tan 0 =

tan 90 =

tan 180 =

tan 270 =

In the trig function , what does the

symbol represent?

How do you solve for if given the

equation

?

Draw and label ASTC. What is the purpose of ASTC?

What is a reference angle? What is the purpose of a reference angle?

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**** Advice for Solving Trigonometry Equations****

1) See if the trig functions “match”

2tanA + √ = tan A versus 2tanA + √ = cot A

2) Substitute a variable in for the “matching” trig functions (optional)

2tanA + √ = tan A

3) Determine if the trig function is positive or negative to see which quadrants you are in (ASTC)

tanA = √ versus tanA = -√

4) In order to find a reference angle, you must perform the inverse of Positive values only!

tanA = -√

6) Be aware of the given interval! (restrictions, degrees or radian measure)

0 versus 180 versus 0

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And, just like the other problems, if the trig function is NOT isolated, isolate it first before you solve for the missing angle. If the problem is given with a domain in terms of , then your answers should be in radians. I suggest doing the problem in degrees first, and then convert to radians.

Model Problem Student Problem

1. Find in the interval that satisfies the equation .

2. Find the value of x in the domain

that satisfies the equation .

3. Find in the interval that satisfies the equation below:

4. Find in the interval that satisfies the equation below:

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Reciprocal Trig Equations

Model Problem Student Problem

5. Find in the interval that satisfies the equation below:

6. Find the value of x in the domain that satisfies the equation below:

Practice:

7. Find , to the nearest tenth of a degree, in the interval that satisfies the equation below:

8. Find in the interval that satisfies the equation below:

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Summary/Closure

Exit Ticket

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Day 1 – HW

9

10

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Day 2: Using Factoring to Solve Trigonometric Equations

SWBAT: solve trigonometric equations by factoring

Warm - Up:

Concept 1: Factorable 2nd degree Trig Equations Each of the following are considered quadratic (2nd degree) trigonometric equations. It should be pretty easy to see why. Algebraic 2

nd Degree Equation Trigonometric 2

nd Degree Equation

Solve for x:

Solve for to the nearest degree in the interval

0o 360

o :

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Example 1: Solve in the interval Factoring Technique:

Example 2: Find all values of x in the interval which satisfies the equation

. Factoring Technique:

Factoring Technique:

To solve a quadratic trig equation:

Factor the quadratic, but instead of using x’s, use “sin x” or whatever function you’re given.

Now you have two linear equations. Solve each of them. You will have anywhere up to 5 solutions!!

Recall that sine x and cosine x can never have a value >1 or <-1. These values will get rejected as solutions.

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4. Factoring Technique:

Practice

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Summary/Closure:

Exit Ticket:

15

Day 2 - HW

16

17

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Day 3 - Solving UnFactorable Trig Equations

SWBAT: Solve trigonometric equations using the quadratic formula

Warm - Up:

1)

2)

19

Algebraic Equation Trigonometric Equation

Example:

( ) √( ) ( )( )

( )

Example:

Find x to the nearest degree in the interval 0o 360o:

( ) √( ) ( )( )

( )

OR

OR

REJECT

Examples:

1. Find to the nearest degree all values of in the interval 0o 360o that satisfies:

4 sin2 – 2 sin – 3 = 0

2. Find to the nearest degree all values of in the interval 0o 360o that satisfies:

9 cos2 – 6 cos = 3

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3.

4. Find to the nearest minute all values of in the interval 0o 360o that satisfies:

4(1 - ) + 5 + 1 = 0.

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Summary/Closure: To solve a trigonometric equation that is not factorable:

Exit Ticket:

22

Day 3 - HW

23

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Day 4: Trig Equations containing more than one function Using Reciprocal Identities

Warm – Up:

Mini – Lesson:

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Let’s Review Reciprocal Identities:

What do you notice about the trig functions below

(“matching”, how to solve, factor, identities etc.)?

Case 1:

Case 2: Case 3:

Technique:

Technique:

Technique:

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Model Problem Student Try It

Example 1: Find all values of A in the

interval 0o 360

o such that

2 sin A - 1 = csc A

Example 2: Find all values of A in the

interval 0o 360

o such that

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Practice: Find all values of x in the interval 0o 360

o such that:

1)

2)

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SUMMARY: Exit Ticket

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Day 4 – Homework

30

31

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Day 5 - Trig Equations containing more than one function USING PYTHAGOREAN IDENTITIES Warm – Up: Match each, but do not solve! (Meaning set up an equation with matching trig functions but do not solve!) 1)

2) Trigonometry Equations: If a trig equation contains more than one function, and the functions cannot be separated out and factored, then you have to convert everything to one equation. One way that this can happen is by using one of the Pythagorean identities. Recall the three Pythagorean Identities: OR OR

We will primarily use only the top two rows.

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Example 1: Find, to the nearest tenth of a degree, all values of in the interval that

satisfy the equation .

Example 2: Find, to the nearest tenth of a degree, all values of in the interval that satisfy the equation

.

34

Example 3: Solve for in the interval 0o 360

o for cos

2 + sin = 1.

Example 4:

35

Summary/Closure

Exit Ticket

36

Day 5 – Homework

37

38

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Day 6 - Solving Trig Equations with Double Angle Identities

Warm-Up:

Examine the following questions below. Write down any observations that you make about the

questions or the trig equations (similarities, differences, how to solve etc.). DO NOT SOLVE!!!!!!

) technique: __________

b) 2 - 1 = technique: __________

c) technique: __________

d) technique: __________

e) technique: __________

40

41

42

43

+ = 0

44

+ = 0

45

46

SUMMARY

Exit Ticket

47

Day 6 – Homework

48

49

50

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