Algebra2/Trig Chapter 13 Packet  White Plains Public ... Chapter 13: Sections 1  Solving First...

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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 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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Table of Contents
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 DoubleAngle 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 = √
5) Use the reference angle to find your answers in the correct quadrants (ASTC)
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:
Set the quadratic = 0, just like you would any quadratic!
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:
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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)
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Quadratics that require the Quadratic Formula
Algebraic Equation Trigonometric Equation
Example:
( ) √( ) ( )( )
( )
√
If asked to the nearest tenthousandth, use your calculator to evaluate:
√
√
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:
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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
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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
.
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Example 3: Solve for in the interval 0o 360
o for cos
2 + sin = 1.
Example 4:
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Summary/Closure
Exit Ticket
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Day 5 – Homework
37
38
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Day 6  Solving Trig Equations with Double Angle Identities
WarmUp:
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
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Day 6 – Homework
48
49
50
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