Download Algebra2/Trig Chapter 12/13 Packet

Algebra2/Trig Chapter 12/13 Packet
In this unit, students will be able to:
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Use the reciprocal trig identities to express any trig function in terms of sine, cosine, or
both.
Derive the three Pythagorean Identities.
Learn and apply the sum and difference identities
Learn and apply the double-angle identities
Learn and apply the ½-angle identities
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: _______
1
Table of Contents
Day 1: Chapter 12-2: Proving Trig identies
SWBAT: Prove trigonometric identities algebraically using a variety of techniques
Pgs. 3 – 9 in Packet
HW:
Pgs. 10 – 12 in Packet
Day 2: Chapter 12-3/12-6: Sum and Difference of Angles Identities
SWBAT: Find trigonometric function values using sum, difference, double, and half angle formulas
Pgs. 13 – 17 in Packet
HW:
Pgs. 18 – 20 in Packet
Day 3: Chapter 13-1: Solving First Degree Trig Equations
SWBAT: Solve First Degree Trig Equations
Pgs. 21 – 25 in Packet
HW:
Pgs. 26 – 27 in Packet
Day 4: Chapter 13-2&3: Solving Second Degree Trig Equations
SWBAT: Solve Second Degree Trig Equations
Pgs. 28 – 32 in Packet
HW:
Pgs. 33 – 34 in Packet
Day 5: Chapter 13-4&5: Solving Trig Equations using Substitution
SWBAT: Solve trigonometric equations using substitution
Pgs. 35 – 39 in Packet
HW:
Pg . 40 in Packet
Day 6
Test Review #1
Pgs. 41 – 42 in Packet
Day 7
Test Review #2
Pgs. 43 – 46 in Packet
2
Make your table here!
3
Chapter 12: Trigonometric Identities
Ch. 12 Section 2 Proving Identities
SWBAT: 1) use the Pythagorean identity to solve for the missing trigonometric function values, 2) rewrite
trigonometric expressions in terms of sine and cosine, and 3) prove identities
A Trig identity is an equation that is true for all values of the variable. You are already familiar with
some of them, even though I have not called them identities before.
Reciprocal Identities
1
csc  
sin 
sec  
1
cos 
cot  
1
tan 
Quotient Identities
sin 
tan  
cos 
cot  
cos 
sin 
Trig Identities are useful in several ways – for simplifying trig expressions, proving other trig identities,
evaluating trig functions, and solving trig equations.
Simplifying Trig Expressions
One way to simplify a complicated trigonometric expression is to rewrite all trig functions in terms of
only sines and cosines.
Simplify each of the following trig expressions by converting them to sines, cosines, or both, and using
your knowledge of fractions, simplify.
Warm - Up:
sec  cot 
=
4
The Pythagorean Identities
Recall, for the unit circle, that
and
.
Also recall that the Pythagorean theorem for any right
triangle is
.
Using
and the Pythagorean theorem, we get
This is the primary Pythagorean Identity.
Please note that from now on, the square of
will be written as
, the square of
will be written as square of
. These are read “sine squared theta” and “cos squared
theta.”
The other two Pythagorean identities are found by dividing the primary identity by
Dividing by
Pythagorean Identity
and
.
Dividing by
Alternative Forms
OR
Basically, whenever you see a “squared,” on one of the trig functions, you should immediately think of
one of the Pythagorean identities.
Example: Use a Pythagorean Identity to express the following expressions in terms of sin , cos , or
both, in simplest form.
c)
a) 1 + cot2 
b) cot2 tan 
5
Proving Trig Identities
To prove an equation is an identity, show that both sides of the equation can be written in the same
form, that is, you see the same thing on both sides, just like when we check equations. To do this, use
valid substitutions and operations and follow the following procedures:
PROCEDURE:
Transform the expression on one side of the equality (usually the more complicated expression) into
the form of the other side (make one side look like the other.) Some strategies for doing this:
Strategy #1: If you see
, and/or the number “1”, use a
Pythagorean identity to change it to something else.
Strategy #2: Convert all functions to sines and cosines. Then try to work with that until lots of stuff
cancels. This might involve complex fractions.
Strategy #3: Put fractions together using a “Fancy Form of 1” or simplify a complex fraction.
Strategy #4: If something is factored, try to distribute. If something can be factored, factor it!
What is difficult about Identity proofs is that knowing what to do is dependent on your experience.
There is often only one or two good ways to go about proving an identity.
Strategy #1: Recognizing Pythagorean Identities
1. Prove:
Strategy #2: Convert all functions to sines and cosines, distribute or factor
2. Prove:
6
Strategy #3: Simplify, combine, or separate fractions, or simplify complex fractions
3. Prove:
4. Prove:
5.
7
6. Prove:
More Practice - Prove the following identities:
1  csc 2  1  sin 

sin 
7. 1  csc 
8. sin A  cos Acot A  csc A
8
Summary/Closure
Exit Ticket:
9
Trig Identities Homework:
Multiple Choice.
1
3
The expression
2
The expression
1)
2)
3)
1)
4)
3)
4)
The expression
1)
2)
3)
4)
is equivalent to
is equivalent to
2)
is equal to
4
The expression
equivalent to
1)
2)
3)
4)
is
10
9
11
Trig Identity Proofs
For all values of for which the
expressions are defined, prove the
identity:
10 For all values of for which the
expressions are defined, prove the
For all values of for which the
expressions are defined, prove that the
following is an identity:
12 For all values of for which the
expressions are defined, prove that the
following is an identity:
identity:
11
13
For all values of for which the
expressions are defined, prove that the
following is an identity:
14 For all values of x for which the
expressions are defined, prove the
following equation is an identity:
12
Chapter 12: 12-3 - 12-6
SWBAT: find trigonometric function values using sum, difference, double, and half angle formulas
Warm - Up
Prove the equation is an identity.
13
Recall that logarithms don’t distribute the way people would THINK they would:
(do you remember what it is?)
Sines and cosines don’t distribute like you think, either. With your calculator, prove that
when A=30 and B=45.
These formulas
WILL be given to
you on the regents
and on my tests.
You just have to
know HOW to use
Concept 1: Sum and Difference of Angles
1. Find the exact value of sin 90 by using the sum of two angles formula
2. Find the exact value of sin 120 by using the difference of two angles formula.
14
3. Find the exact value of cos 75.
4. If sin A = and A is in quadrant II and cos B =
and B is in quadrant I, find cos (A - B).
5. If x and y are acute angles, sin x = and sin y = , then what does sin (x + y) equal in simplest radical
form?
6.
Find the exact value of tan 195.
7.
Find the exact value of tan 15.
15
Concept 1: Double Angles and Half Angles
8. Show that cos 60 = ½ by using cos 2(30).
9. If cos A =
and A is in Quadrant II. Find cos 2A.
10. If cos A = and A is in Quadrant I. Find sin2A.
11. If cos x = , what is the positive value of sin x?
12. If cos A =
5
, find the exact value of tan ½ A.
13
16
Summary/Closure:
Exit Ticket
17
18
19
20
Chapter 13: Sections 1 - Solving First Degree Trigonometric Equations
SWBAT: Solve first degree trig equations
Warm - Up:
If cos A =
 14
64
and A is in QIII, find cos
Algebraic 1st-Degree Equation
1
2
A.
Trigonometric 1st-Degree Equation
2 cos   1  0
2x 1  0
2x  1
2 cos   1
x
cos  
1
2
1
2
Using Unit Circle to identify trig values of quadrantal angles
Draw Unit Circle
Sine
Cosine
Tangent
Sin 0 =
cos 0 =
tan 0 =
Sin 90 =
cos 90 =
tan 90 =
Sin 180 =
cos 180 =
tan 180 =
Sin 270 =
cos 270 =
tan 270 =
21
Concept 1: Trig Equations (sine and cosine) whose results are quadrantal angles.
When you have a trig equation where sine or cosine of the angle = 0,1,-1, you can look at the unit
circle to recognize the values.
1.
2.
3.
4.
5.
6.
If the trig function is not isolated, first you need to isolate the equation, and then you can solve it.
Examples: Solve for
in the domain
.
1.
2.
3.
4.
Concept 2: Trig Equations (sine and cosine) whose results are special angles.
When you have a trig equation where sine or cosine of the angle =
√
,
√
, , you should know that the
reference angle is 30, 45, or 60.
Because all of these values exist in two quadrants for EVERY PROBLEM, there is going to be
more solution. One will be the quadrant I reference angle, and the other will be in either QII, QIII,
or QIV depending on the function used.
You can look at the triangle to find the reference angle that solves the problem, but your calculator
will do it as well, by using either
√
√
1.
2.
22
3.
√
4.
5.
√
If the value is negative, DON’T TYPE THE NEGATIVE in the calculator to find the reference angle!!
Type it in the calculator as if it is positive, find the reference angle, BUT THAT IS NOT A
SOLUTION TO THE PROBLEM!! Use the reference angle to find the actual two solutions in the
two quadrants where that function is negative.
√
3.
√
1.
2.
4.
5.
√
√
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.
1. Find
in the interval
satisfies the equation
that
.
2. If
is a positive acute angle and
, find the number of
degrees in .
23
3. Find the value of x in the domain
that satisfies the equation
.
5.
4. What value of x in the interval
satisfies the equation
√ ?
6.
7.
24
Summary/Closure
Exit Ticket
25
First Degree Trig Equations - HW
26
Answers
27
Chapter 13 Sections 2 and 3: Using Factoring and/or the Quadratic Formula
to Solve Trigonometric Equations
SWBAT: solve trigonometric equations by factoring and/or using the quadratic formula
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
Solve for x:
nd
nd
Degree Equation
Trigonometric 2 Degree Equation
o
Solve for  to the nearest degree in the interval 0 
o
  360 :
Quadratic Equation Warm-up: Solve each of the following quadratic equations by factoring.
a)
b)
c)
28
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.
Example 1: Solve
interval
in the
Example 2: Find all values of x in the interval
which satisfies the equation
.
4.
29
Concept 2: Using the Quadratic Formula to solve difficult-to-factor or
unfactorable 2nd degree Trig Equations
Quadratics that require the Quadratic Formula
Algebraic Equation
Example:
Trigonometric Equation
Example:
Find x to the nearest degree in the interval 0o   
360o:
√
√
√
√
If asked to the nearest ten-thousandth,
use your calculator to evaluate:
and
√
√
√
OR
√
.
OR
REJECT



Examples:
5. Find to the nearest degree all values of  in the interval 0o    360o that satisfies:
4 sin2  – 2 sin  – 3 = 0
6. Find to the nearest degree all values of  in the interval 0o    360o that satisfies:
9 cos2  – 6 cos  = 3
30
7.
31
Summary/Closure: To solve a trigonometric equation that is not factorable:
Exit Ticket:
32
Second – Degree Trig Equations – HW
33
Answer Key:
34
Ch. 13 Sections 4 and 5: Solving by Substitution
SWBAT: solve trigonometric equations using substitution
Warm - Up: Find the exact solution set in the interval 0o
360o for
a)
b) 2 sin2
+ 2 sin
=3
35
Trigonometry: Trig Equations containing more than one function
USING PYTHAGOREAN IDENTITIES
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.
If a trig equation uses more than one function, we’re going to use one of the Pythagorean identities to change
the equation to only have ONE function, then solve as you would have otherwise.
Example 1:
Example 2:
Find, to the nearest tenth of a degree, all
values of in the interval
that
satisfy the equation
.
Find, to the nearest tenth of a degree, all
values of in the interval
that satisfy the equation
.
36
Example 3:
Solve for
in the interval 0o
Example 4: Find all values of A in the interval 0o
360o for cos2
+ sin
= 1.
360o such that
2 sin A + 1 = csc A
Example 5: If 0
2 , find the solution set of the equation 2 sin
= 3 cot .
37
Example 6: Solve for all values of
sin2 - sin
=0
Example 7: Find, to the nearest degree, the roots of:
cos
- 2 cos
=0
38
Summary/Closure
Exit Ticket
39
Solving Trig Equations by Substitution Homework
Answer Key
40
Name:________________________________ Date:______________ Algebra 2 Trigonometry Period:______
Trig Identities and Equations Test Review
1)
2)
3)
4)
5)
41
7)
8)
9)
10) Find all values of
in the interval 0o
360o such that
42
Name:________________________________ Date:________________ A2T Period:_____
Chapters 12 and 13 Trig Identities and Equations Review #2
1)
2)
3) The expression sin 40o cos 10o - cos 40o sin 10o is equivalent to
(A) cos 50o
(B) sin 30o
4) If A is a positive acute angle and sin A =
(A) -
1
3
(B)
1
9
(C) cos 30o
(D) sin 50o
5
, what is cos 2A?
3
(C) -
1
9
(D)
1
3
43
5)
6)
12
3
, cos y = , and x and y are acute angles, the value of cos(x – y) is
13
5
7) If sin x =
(A)
21
65
8) If cos x =
(B) -
33
65
(C) -
14
65
(D)
63
65
and x is in the second quadrant, find the exact value of sin x.
44
9)
10)
11)
45
12)
13)
46