Chapter 5 Trigonometric Functions - Glendale …web.gccaz.edu/.../HoltfrerichHaughnPreCalculusChapter6.doc · Web viewIn this section you will get a chance to practice your problem

Chapter 6 Trigonometric Identities and Equations (picture of someone tuning a piano)

Ever wonder how a piano tuner does his or her job? The quick answer is that they listen for lower

beat frequencies created by what is being tuned and a tuning fork, and they try eliminating that sound

by adjusting the instrument being tuned. The long answer is that there is a lot of math behind what is

going on. If two instruments are making sound at the same time their sound waves are being added.

In math terms the combined amplitude (size of the sound) is represented by:

By using the trigonometric identity , that you will be learning

about in this chapter, we can alter the expression above into something that is equivalent to A.

The factor of the expression represents the sound whose frequency is the average

of the two instruments and the factor represents the lower beat frequency that the

tuner is trying to eliminate. By the use of a trigonometric identity one can show what is going on and

why, what the tuner is doing, works!

Chapter 6 Trigonometric Identities and Equations

Section 6.1 Trigonometric Identities

Objectives

Understanding the basic identities

Using identities to simplify

In this chapter we are going to look at a lot of identities used to manipulate trigonometric

expressions. As you learn to solve equations and evaluate trigonometric functions you will found that

you will need to change trigonometric functions by using identities in order to be capable of

answering questions.

THE BASIC IDENTITIES

You have seen some of the identities we will be looking at in this chapter. An identity is something

which is true for all inputs. For example, x + 7 = 7 + x is an example of an identity. It is true for all

real numbers. Here is a list of some of the basic trigonometric identities.

Reciprocal Identities

csc u = sec u = cot u =

sin u = cos u = tan u =

Quotient Identities

tan u = cot u =

Pythagorean Identities

Cofunction Identities

You did see all of these identities in chapter 5 but they just weren’t specifically pointed out as such.

They come out of the definitions of the trigonometric functions. Let’s look at where a few of these

come from.

Discussion 1: Proof of Some Identities


Let’s investigate how we get some of our identities.

Unit circle definitions of trigonometric

functions., , , the other three

are just the reciprocals of these three.

Let’s look at the quotient identity for tangent.by definition.

Since and we substitute and

get; .

Let’s see where the Pythagorean identities

come from. Remember that with the unit

circle you have a right triangle. If you look at

the triangle notice that . Therefore,

use substitution again and we have the first

identity

, ,

so, which is written as;

To get the other two Pythagorean identities

simply divide the first identity found by

cos2u and then by sin2u. Notice that

is only true if cos2u 0, and

is only true if sin2u 0.

This is a start at showing you how mathematicians use the definitions and identities to arrive at new

identities and formulas.

Example 1: Two Ways to Answer

What are two ways of answering the question of finding sin θ given that cos θ = and knowing

that θ is a third quadrant angle?

Solution:

Method 1: Make a triangle and use the

coordinate definitions (used in section 5.5

example 6).

, 52 − (4)2 = y2 y2 = 25 − 16 = 9 y = 3

So, for this example y = −3 since we go down

to the point in the third quadrant. Our answer

1

xu

P (x, y)

(1, 0)

y

θ

y = ?r = 5

x = −4


then is,

Method 2: Use the Pythagorean identity

, and substitute in the

value we know for . Using this

method does leave us at the end wondering

if the answer is positive or negative. But, if

we have memorized ASTC we know that

only tangent and cotangent are positive in

the third quadrant thus sine must be

negative in this example.

Question 1: What could cos θ be if given that sin θ = ?

Example 2: Finding Values Using Identities

What are the values of all six trigonometric functions given that cos θ = and csc θ = ?

Use identities to calculate your answers.

Solution:

We need to think about

which identities will help us.

Since we have csc θ we could

use the reciprocal identity to

find sin θ. We would then

know sine and cosine so we

could use the quotient

identity to find tan θ. And

then we could use the

reciprocal identities to find

cot θ and sec θ.

sin u = , tan u = , cot u = , sec u =

cos θ = , csc θ = , so

SIMPLIFING WITH IDENTITIES


Discussion 2: Simplifying ExpressionsSimplify the following expression by using identities.

a) cot x sin x b) cos 2 x (sec 2 x − 1) c) (1 − sin 2 x) sec 2 x

a) Convert cot x into sines and cosines by a

quotient identity.cot x sin x = = cos x

b) Convert (sec 2 x − 1) into tan 2 x by a

Pythagorean identity. Then, convert

tangent by using a quotient identity.

cos 2 x (sec 2 x − 1) = cos 2 x tan 2 x

=

= sin 2 x

c) Convert (1 − sin 2 x) into cos 2 x by a

Pythagorean identity. Then, convert

secant by using a reciprocal identity.

(1 − sin 2 x) sec 2 x = cos 2 x sec 2 x

=

= 1

Question 2: Can be simplified, and if so how?

In the last question you might have been tempted to replace (1 − cos 2 x) with sin 2 x but that doesn’t

lead you anywhere. Factoring the numerator as a difference of two squares is the key. We can’t

forget our algebra. We need it here in trigonometry too.

Example 3: Simplify Expressions

What are the simplified versions of the following?

a) b) sec cos θ c) sec 2 x tan 2 x + sec 2 x

Solution:

a) Use a Pythagorean identity then

reciprocal identity= = tan 2 x

b) Use a cofunction identity then reciprocal

then quotient.sec cos θ = csc θ cos θ

= cos θ

=

c) This time we need to factor then use a

Pythagorean identity.

sec 2 x tan 2 x + sec 2 x = sec 2 x (tan 2 x + 1)

= sec 2 x sec 2 x


= sec 4 x

Section Summary:

Be sure to memorize your identities.

Reciprocal Identities

csc u = sec u = cot u =

sin u = cos u = tan u =

Quotient Identities

tan u = cot u =

Pythagorean Identities

Cofunction Identities

Answer Q1:

,

which answer will depend on whether θ is in the third or fourth quadrants.


SECTION 6.1 PRACTICE SET(1-- 4) Prove the following identities using the unit circle definition.

1. 2.

3. 4.

(530) (If no answers are possible explain why)

5. Given in the 1st quadrant find the other five trigonometric function values.


7. Given in the 3rd quadrant find the other five trigonometric function values.

8. Given in the 4th quadrant find the other five trigonometric function values.



11. Given in the 2nd quadrant find the other five trigonometric function values.


13. in the 1st quadrant find the other five trigonometric function values.




Answer Q2:

=

=The difference of two squares.


17. Given in the second quadrant find the other five trigonometric function values.











28. Given in the second quadrant find the other five trigonometric function values.



(3136) Using trigonometric identities find the values of the other four trigonometric functions given the values of two of the trigonometric functions.

31. 32.


33. 34.

35. 36.

(3758) Simplify each of the following:

37. 38. 39.

40. 41. 42.

43. 44. 45.

46. 47. 48.

49. 50. 51.

52. 53. 54.

55. 56. 57.

58.


Section 6.2 Verifying Identities

Objectives

Understanding how to verify identities

In this section you will get a chance to practice your problem solving skills and you will get a better

understanding of how the trigonometric functions relate to each other. Verifying identities will

require that you know all of your identities well. When you verify an identity you are checking to see

if two different expressions are equivalent.

Caution, this is different from solving equations. You do not want to move terms from one side of

the equal sign to the other or multiply both sides of the equation by something. You must work with

only one side of the equation at a time.

IDENTITY VERIFICATION

Let’s begin with an example.

Discussion 1: Verifying Identities

Verify the identity .

We might begin by working with the left side since it is

the more complicated looking of the two. When verifying

identities there can be different ways of proving them

equivalent.

One way of starting this example might be to change the

sec 2 x − 1 by using a Pythagorean identity.

Next we might eliminate the fraction by changing sec 2 x

by using a reciprocal identity.

Now, change tan 2 x into sines and cosines and simplify

(quotient identity).

This isn’t the only way to approach this example. Here is another way to work the problem.

As above we begin by working with the left side since it is

the more complicated looking one of the two.


We now break the fraction up into two fractions.

Next we simplify each fraction.

Now, convert , by using a Pythagorean identity.

Let’s talk about some guidelines that can be helpful when trying to verify identities.

1) You must have your basic identities memorized!

2) You should work with the more complicated looking side first. Remember that you

can’t move terms from one side to the other or multiply both sides by something.

3) Typically, you will want to add fractions together, simplify fractions so that they have

monomials in the denominator, and/or factor when possible.

4) Look for opportunities to use trigonometric identities to get functions that are the same

or that are paired up like sine and cosine, or tangent and secant, or cotangent and

cosecant or that are paired up with the other side of the identity.

5) Another strategy might be to convert everything to sines and cosines.

6) You may want to multiply the numerators and denominators of fractions by something

in order to create the difference of two squares like multiplying (1 + sin x) by (1 − sin x)

to get (1 − sin 2 x) which equals cos 2 x.

7) If nothing comes to mind just try something. It may lead somewhere or it might not but

either way you will gain some insight about how to verify the identity.

Question 1: What would you want to try first when verifying ?

Example 1: Verifying IdentitiesVerify that the following are identities.

a) b) c)

Solution:

a) Here we would like to change tan 2 x + 1 so that

everything will be in secants.

Now canceling out the like factor secant, and

you are done.

b) Here it is hard to say which is more complicated

but let’s try putting the right side into one term


by subtracting them. First, we would need to

change secant by the reciprocal identity.

Next, get common denominators and subtract.

Now convert the numerator by a Pythagorean

identity.

c) On this problem let’s try manipulating the left

side. We should get a single term in the

denominator by replacing (1 + cot2 t) with csc2t.

Now let’s use a reciprocal identity to change

csc 2 t.

Next, distribute and replace cot 2 t with .

The sin 2t canceled out

Lastly use a Pythagorean identity to convert

cos 2 t to (1 − sin 2 t) and then simplify.

Example 2: Working with both sides

Verify that is an identity.

Solution:

It might be easier if we change all

trigonometric functions to sines and

cosines.

Thus, replace csc θ with .

Next, let’s work with the left side.

Let’s use hint number 6.

Combine the two fractions.

Simplify.

Factor and replace (1−cos2θ) with

csc θ(1 + cos 2 θ)


sin 2 θ.

Now cancel the sines.

Here is an example of one area where trigonometric functions show up in electronics.

Example 3: Capacitors and InductorsThe energy stored in a capacitor is given by the function C(t) = k cos 2 (2πFt) and the energy stored in

an inductor is given by I(t) = k sin 2 (2πFt) where t is time and F is the frequency. The total energy in

a circuit is given by E(t) = C(t) + I(t). Prove that E(t) must be a constant.

Solution:

Let’s begin by substituting into the formula for E(t).

Now factor out the k.

cos 2 (u) + sin 2 (u ) = 1, so;

E(t) = C(t) + I(t)

E(t) = k cos 2 (2πFt) + k sin 2 (2πFt)

E(t) = k [cos 2 (2πFt) + sin 2 (2πFt)]

E(t) = k (1) = k a constant.

Section Summary:

Hints for verifying identities:

1) You must have your basic identities memorized!

2) You should work with the more complicated looking side first. Remember that you

can’t move terms from one side to the other or multiply both sides by something.

3) Typically, you will want to add fractions together, simplify fractions so that they have

monomials in the denominator, and factor when possible.

4) Look for opportunities to use trigonometric identities to get functions that are the same

or that are paired up like sine and cosine, or tangent and secant, or cotangent and

cosecant or that are paired up with the other side of the identity.

5) Another strategy might be to convert everything to sines and cosines.

6) You may want to multiply the numerators and denominators of fractions by something

in order to create the difference of two squares like multiplying (1 + sin x) by (1 − sin x)

to get (1 − sin 2 x) which equals cos 2 x.

7) If nothing comes to mind just try something. It may lead somewhere or it might not but

either way you will gain some insight about how to verify the identity.

Answer Q1:You would want to change tan 2x + 1 into sec2x.


SECTION 6.2 PRACTICE SET(1– 64) Prove the following identities are true:

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

29. 30.

31. 32.


33. 34.

35. 36.

37. 38.

39. 40.

41. 42.

43. 44.

45. 46.

47. 48.

49. 50.

51. 52.

53. 54.

55. 56.

57. 58.

59. 60.

61. 62.

63. 64.

(65−70) Graph each of the following to show they are identities, then prove they are identities.


65. 66.

67. 68.

69. 70.

(71−76) Show each of the following are not identities by finding an angle value for that makes the statement false. (Check by using the graphing calculator)

71. 72.

73. 74.

75. 76.

(77−82) Show each of the following are not true by finding an angle values for x and y so the statement is false.

77. 78.

79. 80.

81. 82.


Section 6.3 More Identities

Objectives

Understanding more complicated identities

In this section we are going to look at more trigonometric identities. Some are used more frequently

than others, but you will need them as you study more mathematics.

MORE COMPLICATED IDENTITIES

First we will begin with identities involving the addition and subtraction of angles.

Sum and Difference Identities

Let’s take some time here to prove one of the above identities.

Since arc AC is equivalent to arc BD (both arcs have the same

angle (u − v)), line segment is equal in length to .

Therefore, we can create the following equation and result.

Distance formula

Square both sides

Square terms

sin2x+cos2x=1

subtract 2

divide by −2

The other formulas can be proven as well but we are not going to do that here. Here is an example of

one way in which these were used before calculators came on the scene.

Example 1: Finding Trigonometric Values

u−vv

u

B(cos v, sin v)

A(1, 0)D(cos u, sin u)

C(cos (u−v), sin (u−v))


Find the cos .

Solution:

We need to convert this to angles we know

like , , , and any multiple of these.

Now, we can use a difference identity and

determine the value of cos .

cos = cos

cos =

=

=

Example 2: Algebraic Use

What would simplify too? Notice that this example fits here because inverse

trigonometric functions yield angle answers. This is the sine of the sum of two angles.

Solution:

Sum identity for sine.

We now have:

Simplify the inverses.

Now we need a triangle to

help us with the other two

parts. Thus, =

sin θ = and

= cos α

= .

Perform the multiplication.

Simplify.

cos −1 x = θ sin −1 x = α

Here are some more identities that show up in calculus and other courses.

θ

y = r = 1

x = x

αy = x

r = 1

x =


Double Angle and Power Reducing Identities

Discussion 1: Proving Some IdentitiesLet’s show how some of the above identities are derived.

Let’s start by looking at sin 2u. We will

verify the identity.

First, let’s rewrite the left side.

Second, use the sum identity.

Lastly, combine like terms.

cos 2u is verified similarly, so let’s assume

it’s an identity and verify tan 2u.

First, let’s replace tan 2u by a quotient

identity.

Now, substitute with a double angle

identities.

Divide the numerator and the denominator

by cos 2 u.

Simplify.

Lastly,

Let’s do one more and verify the identity

.

To start we will use the cos 2u identity on

the right side.


Now, replace sin 2 u with a Pythagorean

identity.

Simplify the numerator.

Finally, we are done.

Question 1: Verify the identity .

Example 3: Simplifying a Double Angle

Find the following. cos (2cos −1 x)

Solution:

Double angle identity.

We now have:

Simplify the inverses.

Now we need a triangle to

help us with, =

sin θ = .

Perform the multiplication.

Simplify.

cos −1 x = θ

Of course, we have more identities. Let’s look at what are called half angle identities.

Question 2: What do you think could be the right side of this identity sin = _______?

(Hint: The power reduction formula is a good start.)

θ

y = r = 1

x = x


Discussion 2: Discovering the Identity for

Let’s look at how we could have answered question 2.

We will start with the power reduction

identity for sin 2 u.

Let’s take the square root of both sides.

We’re close, now just replace u with .

Simplify and we’re done.

The other half angle identities can be verified easily. Here are all three of them.

Half Angle Identities

With the sine or cosine of half an angle, the + or − symbol is determined by which quadrant the angle

is located. Here are a couple of examples.

Example 4: Using the Half Angle Identities

What are the values of sin , and cos exactly?

Solution:

The sin = sin which means that

the half angle identity can help us.

We evaluate cos .

Simplify the square root.


The final result is positive because is

in the first quadrant where sine is positive.

The cos = cos which means

that the half angle identity can help us.

We evaluate cos .

Simplify the square root.

The final result is negative because is

in the second quadrant where cosine is

negative.

Question 3: Why did the two answers to example 4 differ by only a negative sign?

Example 5: Double and Half Angles

Given sin θ = and that < θ < , find sin 2θ, cos 2θ, , and .

Solution:

For finding sin 2θ, we will use a

double angle formula along with a

picture of what is given and our

knowledge that θ is in the third

quadrant. sin 2θ = 2 sin θ cos θ

θy = −3r = 5

x = 4

Answer Q1:

Answer Q2:


= 2 =

As in the first part we will find cos 2θ cos 2θ

= 1 2

= 1 =

Now we find . Since θ is in

the third quadrant half of θ will be in

either the first or second quadrants.

Sine yields positive values in those

two quadrants.

(sign depends on quadrant)

= =

= or

As with we find .

Since the reference angle in quadrant

three is less than 45° (the magnitude

of x is bigger than the magnitude of y)

half of θ will put the angle in the first

quadrant where cosine yields positive

values.

= =

= or

Lastly, we have these identities.

Product to Sum

Sum to Product

Example 6: Using Product Sum Identities

Convert the following products to sums and sums to products.


a) sin θ cos 2θ b) cos 3θ + cos θ

Solution:

a) We need the formula for the product of

sine and cosine.

Substitute in our angles.

Simplify the right side.

Sine is an odd function.

b) We need the formula for the sum of two

cosines

Substitute in our angles.

Simplify the right side.

Fractions simplify.

Example 7: Using Identities

Simplify the following by using an identity.

a) b)

Solution:

a) This looks like one of the sum or

difference identities. In fact it is the

cosine of the difference of two angles,

thus we get:

b) This looks like one of the sum to product

identities. In fact it is the cosine minus

cosine, thus we get:

Section Summary:

Identities:

Sum and Difference Identities

Answer Q3:The angles

and

are

complementary.

Thus sin

and cos

are equal. Now

has a

reference angle

of so

that is why we get the same answers just opposite signs.


Double Angle and Power Reducing Identities

Half Angle Identities

Product to Sum

Sum to Product


SECTION 6.3 PRACTICE SET(1–12) Use the sum and difference identities to evaluate exactly the following:

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

(13–26) Show each of the following is true using sum and difference identities.

13. 14. 15.

16. 17. 18.

19. 20. 21.

22. 23. 24.

25. 26.

(27–36) Use the sum and difference identities in reverse to rewrite each of the following as a single trigonometric function.

27. 28.

29. 30.

31. 32.

33. 34.

35. 36.


(37–44) Use the given information to find exact values of:

a.) sin 2 b.) cos 2 c.) d.)

37. 38.

39. 40.

41. 42.

43. 44.

(45–48) Use the given information to find exact values of:

a.) b.

45. 46.

47. 48.

(49–70) Prove the following identities.

49. 50.

51. 52.

53. 54.

55. 56.

57. 58.

59. 60.

61. 62.

63. 64.


65. 66.

67. 68.

69. 70.

(71–76) Convert the following products to sums or differences.

71. 72. 73.

74. 75. 76.

(77–84) Convert the following sums or differences to products.

77. 78. 80.

80. 81. 82.

83. 84.


Section 6.4 Solving Trigonometric Equations

Objectives

Understanding how to solve trigonometric equations

It is now time to solve equations that have trigonometric functions. To solve equations of this

type you will need to get the equations into factors equal to zero with each factor containing

only one trigonometric function.

General Strategy:

1. Get the equation equal to zero

2. Convert all trigonometric functions into the same function by using identities or if

that isn’t possible, then factor the equation into factors where each has only one

type of trigonometric function.

3. Set each factor equal to zero and solve for the trigonometric function.

4. Lastly identify which angles make each equation true.

SOLVING TRIGONOMETRIC EQUATIONS

Discussion 1: Solving a Simple Equation

Let’s find all the angles that make 2sin θ − 1 = 0.

As we think about our strategy for solving this we see that it is equal to zero already and that there is

only one function.

Therefore, we will begin by getting sin θ

alone.

2sin θ − 1 = 0

2sin θ = 1 (added one)

sin θ = (divided by two)

Now, we need to think about what angle

input would cause the output of the sine

function to be .

sin θ = y on the unit circle. An angle that has a

point on the unit circle with y-value of is .

We know that is one angle. Now we

need to think about every other angle

whose reference angle is that would

cause the output of the sine function to

be positive.

Sine is positive in the 1st and 2nd quadrants. So,

we have and (whose reference angle in the

2nd quadrant is ). In addition to these two we

also have every 2π revolutions around the unit

circle from these two angles, which will also


make sine equal , since sine is periodic with a

period of 2π.

Final answer.θ = and θ = (n any integer)

We could have used the calculator to help us with this problem as long as we are

comfortable with non-exact solutions.We would still need to solve for sine. 2sin θ − 1 = 0

2sin θ = 1 (added one)

sin θ = (divided by two)

But now we could use the calculator and

find the sin -1 =θ.

Notice that 0.52359… is the same as which we found earlier as one of the many answers.

But since we are using the inverse sine function to assist us in solving our problem we only

get one answer on the calculator even though there are many answers to sin θ = . We still

need to think in order to arrive at all of the solutions to the equation. The answer 0.52359…

is in radians and is the reference angle so in the second quadrant the angle would be π

0.52359… = 2.61799…

Final answer. θ = and θ = (n any integer)

Remember that the trigonometric functions are not one-to-one and thus we have many inputs that

yield the same outputs. Specifically we have an infinite number of them. In “real life” though, it may

very well be that all we need are the answers that exist in just the first revolution (in the domain of

).

Question 1: If we have cos x = what angles, x, would make this equation true?

Let’s look at some more examples.

Example 1: Solving Equations

Solve the following equations.

a) b) c)


Solution:

a) Get the equation equal to zero

Factor

Set each factor equal to zero and solve.

We now look for angles that will satisfy

each equation on the domain of ,

then we extend to the whole domain.

Notice the πn in both answers. This happened because our two answers in each case were

half a revolution apart, so we could combine them into just one. To finally answer the

question, what is the solution to , we need to make one more

observation. Tan θ is not defined for , so our final solution is .

b) This is already set equal to zero, so we

begin by factoring.

Now set each factor equal to zero and

solve.


each equation on the domain of ,


Now the final answer.

is impossible so there isn’t a

solution from this equation. With the

second, think what makes (secant

and cosine are reciprocals). Two angles that

work are and .

x =

c) Get the equation equal to zero

Use a Pythagorean identity to convert

cosecant squared to cotangent squared.


Solve for cotangent.

A more convenient way to write

when looking for exact answers.


this equation on the domain of ,


Now the final answer.

It might be easiest to think what angles

cause tangent to equal .

Let’s look at another example.

Discussion 2: Multiple of an Angle

Let’s find the solution, on the domain of , for .

Just as with our previous equations we

need to get the sine alone.

Next, we need to think what angles yield

as an output (write all the answers).

Notice that we are not concerned about

the 3x at the moment.

(θ = 3x)

Answer Q1:The reference angle that has

as an

answer is .

Since cosine is negative we want angles in the 2nd and 3rd quadrants. Thus

x =

or

x =


Now we can solve for x.

(multiplied by )

To get our final answer remember that

we were looking for answers on the

domain of . Since this function,

sin 3x, has a period that is the usual

period we need to look for 3 times as

many answers as usual.

The final answer is…

It is easy to forget to find all the answers to an equation like the one we just solved. Make sure that

once you think you have finished a problem check to see that you have found all of the possible

solutions. Let’s do another example.

Example 2: Fraction of an angle

Find all of the solutions to on the domain of .

Solution:

It looks as though we will need to

factor and then set each factor

equal to zero.

Set each factor equal to zero and

solve for the trigonometric

function.

As was done in discussion 2,

think what angles will make each

of our two equations true.

(θ = , )

Now we can solve for x.

To get our final answer remember

that we were looking for answers

on the domain of .


Section Summary:Here are the guidelines for solving trigonometric equations.

1. Get the equation equal to zero

2. Convert all trigonometric functions into the same function by using identities or if that isn’t

possible, then factor the equation into factors where each has only one type of trigonometric

function.

3. Set each factor equal to zero and solve for the trigonometric function.

4. Lastly identify which angles make each equation true.

Remember that when you are solving equations where the trigonometric functions have multiples of

angles we need to be very careful about finding all of the solutions.


SECTION 6.4 PRACTICE SET(1–22) Solve each equation for , with 0<<2. (Give the answers in radians)

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14. 15.

16. 17. 18.

19. 20. 21.

22.

(23–44) Solve each equation for , with 0<<360o. (Give the answers in degrees)

23. 24. 25.

26. 27. 28.

29. 30. 31.

32. 33. 34.

35. 36. 37.

38. 39. 40.

41. 42. 43.

44.


(45–54) Each of the following has an infinite number of solutions, write an expression the represents all the possible solutions. (Give the expression in radians)

45. 46. 47.

48. 49. 50.

51. 52. 53.

54.

(55–64) Use the calculator to solve each equation for , with 0<<2. (Give the answer in radians, approximated to two decimal places, if there is no answer explain why)

55. 56. 57.

58. 59. 60.

61. 62.

63. (Hint: Use the quadratic formula)


(65–70) Use the calculator to solve each equation for , with 0<<360o. (Give the answer in degrees, approximated to two decimal places, if there is no answer explain why)

65. 66. 67.

68. 69. (Hint: Use the quadratic formula)


(71–76) Graph each of the following with the graphing calculator and use the graphing calculator to find all solution(s) for , with 0<x<2. (Give the answer(s) in radians approximated to two decimal places, if there is no solution explain how the graph tells you this)

71. 72. 73.

74. 75. 76.

77. 78.

(79– ) Mathematical modeling needed


Chapter 6 ReviewTopic Section Key Points

CHAPTER 6 REVIEW HOMEWORK

Section 6.1

1. Given is in the 1st quadrant, find the other five trigonometric functions.

2. Given is in the 2nd quadrant, find the other five trigonometric functions.

3. Given is in the 3rd quadrant, find the other five trigonometric functions.

4. Given is in the 4th quadrant, find the other five trigonometric functions.

(5−6) Given the value of two trigonometric functions, use the trigonometric identities to find the values of the other four trigonometric functions.

5.

6.

(710) Simplify each of the following:

7. 8.

9. 10.

Section 6.2

(1118) Prove the following identities are true.

11. 12.

13. 14.

15. 16.

17. 18.

(1920) Show each of the following are not identities by finding an angle that makes the statement false.

Section 5.1 Geometry Review

19. 20.

(2122) Graph each of the following to show they are identities, then prove they are identities.

21. 22.

Section 6.3

(2326) Use the sum and difference identities to evaluate exactly the following:

23. 24.

25. 26.

(2730) Use the sum and difference identities in reverse to rewrite each of the following as a single trigonometric function.

27. 28.

29. 30.

(3134) Use the given information to find the exact values of:

31. 32.

33. 34.

(3536) Use the given information to find the exact values of:

35. 36.

(3746) Prove the following identities.

37. 38.

39. 40.

Chapter 5 Trigonometric Functions

41. 42.

43. 44.

45. 46.

(4748) Convert the following products to sums or differences.

47. 48.

(4952) Convert the following sums or differences to products.

49. 50.

51. 52.

Section 6.4

(5360) Solve each equation for , with 0< < 2. (Give the answers in radians)

53. 54.

55. 56.

57. 58.

59. 60.

(6168) Solve each equation for , with 0< < 360o. (Given the answers in degrees)

61. 62.

63. 64.

65. 66.

67. 68.

(6972) Each of the following has an infinite number of solutions, write an expression that represents all the possible solutions. (Given the answer in radians)

69. 70.


71. 72.

(7378) Use the calculator to solve each equation for , with 0< < 2. (Give the answers in radians approximated to 2 decimal places)

73. 74.

75. 76.

77. 78.

(7984) Use the calculator to solve each equation for . with 0< < 360o/ (Give the answers in degrees approximated to 2 decimal places)

79. 80.

81. 82.

83. 84.

(8588) Graph each of the following with the graphing calculator and use the graphing calculator to find all the solutions of x, with 0 < x < 2. (Give the answer(s) in degrees approximated to 2 decimal places , if there is no solutions explain how the graph tells you this)

85. 86.

87. 88.

Chapter 5 Trigonometric Functions

CHAPTER 6 EXAM

1. Using the unit circle definition prove:

2. Given in the 2nd quadrant, find the other five trigonometric function values.

3. Given find the values of the other four trigonometric functions.

(45) Simplify:

4. 5.

(613) Prove the following identities are true

6. 7.

8. 9.

10. 11.

12. 13.

(1416) Use the sum and difference identities in reverse to rewrite each of the following as a single trigonometric function.

14. 15.

16.

(1719) Use the sum and difference identities to evaluate each of the following exactly.

17. 18. 19.

(2122) Use the given information to find exact values of:

21. 22.


23. Given find:

(2425) Convert the following to a sum or difference.

24. 25.

(2627) Convert the following sums to a product.

26. 27.

(2833) Solve each of the following equations for , with . (Given the answers in radians and degrees)

28. 29.

30. 31.

32. 33.

(3435) Use the graphing calculator to solve each equation for , with . (Give the answers in radians, approximated to two decimal places)

34. 35.

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