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Algebra2/Trig Chapter 12/13 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.

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: _______

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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

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Make your table here!

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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 Quotient Identities

1csc

sin

sintan

cos

1sec

cos

coscot

sin

1cot

tan

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 =

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The Pythagorean Identities

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 and .

Dividing by

Dividing by

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.

a) 1 + cot2

b) cot2 tan

c)

Pythagorean Identity Alternative Forms

OR

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.

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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:

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Strategy #3: Simplify, combine, or separate fractions, or simplify complex fractions

3. Prove:

4. Prove:

5.

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6. Prove:

More Practice - Prove the following identities:

7.

21 csc 1 sin

1 csc sin

8. sin cos cot cscA A A A

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Summary/Closure

Exit Ticket:

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Trig Identities Homework:

Multiple Choice.

1 The expression is equivalent to

1) 2) 3)

4)

2 The expression is equivalent to

1)

2)

3) 4)

3 The expression is equal to

1) 2) 3) 4)

4 The expression is

equivalent to 1) 2) 3) 4)

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Trig Identity Proofs

9 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

identity:

11 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:

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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:

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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.

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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.

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.

These formulas WILL be given to

you on the regents and on my tests. You just have to

know HOW to use

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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.

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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 = 13

5, find the exact value of tan ½ A.

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Summary/Closure:

Exit Ticket

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Chapter 13: Sections 1 - Solving First Degree Trigonometric Equations

SWBAT: Solve first degree trig equations

Warm - Up:

If cos A = 1464

and A is in QIII, find cos 12

A .

Algebraic 1st-Degree Equation Trigonometric 1st-Degree Equation

12

2 1 0

2 1

x

x

x

12

2 cos 1 0

2 cos 1

cos

Using Unit Circle to identify trig values of quadrantal angles Draw Unit Circle

Sine

Cosine

Tangent

Sin 0 =

Sin 90 =

Sin 180 =

Sin 270 =

cos 0 =

cos 90 =

cos 180 =

cos 270 =

tan 0 =

tan 90 =

tan 180 =

tan 270 =

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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.

√

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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.

√

1.

2.

√

3. √

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 that satisfies the equation .

2. If is a positive acute angle and , find the number of

degrees in .

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3. Find the value of x in the domain

that satisfies the equation

.

4. What value of x in the interval

satisfies the equation

√ ?

5. 6.

7.

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Summary/Closure

Exit Ticket

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First Degree Trig Equations - HW

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Answers

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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

nd Degree Equation Trigonometric 2

nd Degree Equation

Solve for x:

Solve for to the nearest degree in the interval 0o

360o :

Quadratic Equation Warm-up: Solve each of the following quadratic equations by factoring.

a)

b)

c)

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Example 1: Solve in the interval

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

.

4.

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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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 Trigonometric Equation

Example:

√

√

If asked to the nearest ten-thousandth,

use your calculator to evaluate: √

and √

.

Example:

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

√

√

√

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

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7.

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

Exit Ticket:

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Second – Degree Trig Equations – HW

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Answer Key:

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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 360

o for

a)

b) 2 sin2 + 2 sin = 3

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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: 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: Find all values of A in the interval 0o 360

o such that

2 sin A + 1 = csc A

Example 5: If 0 2 , find the solution set of the equation 2 sin = 3 cot .

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Example 6: Solve for all values of sin2 - sin = 0

Example 7: Find, to the nearest degree, the roots of: cos - 2 cos = 0

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Summary/Closure

Exit Ticket

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Solving Trig Equations by Substitution Homework

Answer Key

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Name:________________________________ Date:______________ Algebra 2 Trigonometry Period:______

Trig Identities and Equations Test Review

1)

2)

3)

4)

5)

42

7)

8)

9)

10) Find all values of in the interval 0o 360

o such that

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Name:________________________________ Date:________________ A2T Period:_____

Chapters 12 and 13 Trig Identities and Equations Review #2

1)

2)

3) The expression sin 40o

cos 10o - cos 40

o sin 10

o is equivalent to

(A) cos 50o (B) sin 30

o (C) cos 30

o (D) sin 50

o

4) If A is a positive acute angle and sin A = 3

5, what is cos 2A?

(A) -3

1 (B)

9

1 (C) -

9

1 (D)

3

1

44

5)

6)

7) If sin x = 13

12, cos y =

5

3, and x and y are acute angles, the value of cos(x – y) is

(A) 65

21 (B) -

65

33 (C) -

65

14 (D)

65

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8) If cos x =

and x is in the second quadrant, find the exact value of sin x.

45

9)

10)

11)

46

12)

13)