Lesson #42 Three Basic Trigonometric Functions€¦ · trigonometric functions because they are reciprocals of the three original trigonometric ... Sine and cosecant are reciprocals

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Lesson #42 – Three Basic Trigonometric Functions A2.A.66 Determine the trigonometric functions of any angle, using technology

A2.A.64 Use inverse functions to find the measure of an angle, given its sine,

cosine, or tangent

Term Formula/Symbol Important Information

Theta A variable used in trigonometry to represent an

unknown angle. It is not like π because π has a

value (approximately 3.14). Think of it like using

x, y, t, or any other variable.

Pythagorean

Theorem:

2 2 2a b c

This can only be used with right triangles!

Sine of an angle sin( )opposite

xhypotenuse

In a right triangle, the ratio between the side

Opposite an angle and the Hypotenuse

Cosine of an angle cos( )

adjacenty

hypotenuse In a right triangle, the ratio between the side

Adjacent or next to an angle and the Hypotenuse

Tangent of an angle tanopposite

adjacent In a right triangle, the ratio between the side

Opposite an angle and the side Adjacent to the

angle

Memory Device:

Note: There is a lot more to the definitions of sine, cosine, and tangent. We will be

expanding our understanding of these trig. functions throughout the unit.

We read sin(x) as “sine of x” and not “sine times x” because SINE IS A FUNCTION OF X. The

sine notation, sin(x) is the same as function notation f(x). Sine is the specific name of a

function where your input is an angle and your output is the ratio between the side opposite

that angle and the hypotenuse in a right triangle.

Make sure your calculator is in degree mode! Evaluate each of the following trigonometric functions to the nearest ten thousandth.

1. cos (73°)=

2. sin (73°)=

3. tan 81°=

4. sin 59 =

5. sin 33 =

6. tan 245 =

~ 2 ~

7. f(x) = tan(x).

Find f(38.9 ).

8. f() = tan().

Find f(57 ).

9. g(x) = cos(x).

Find g(-100 ).

How are questions 6 and 9 possible when those angles would not fit in a right triangle?

Obviously there is more to sine, cosine, and tangent than what you know so far.

The inverse trigonometric functions We learned that all one-to-one functions also have inverses that are

functions. (For now, we won’t worry about the 1-to-1 part.)

The inverse function for y=sin(x) is y= 1sin ( )x .

Similarly, the inverse function for y=cos(x) is __________.

And the inverse function for y=tan(x) is ___________.

The input (x) for a trig function is the angle, and the output (y) is the trigonometric ratio.

Therefore the input for an inverse trigonometric function is the ___________________ and

the output is the ___________.

Evaluate each of the following inverse trigonometric functions to the nearest degree.

1 1sin

2

1cos .6789 1tan 3

Solve for the variable to the nearest degree.

10. Solve for : sin = .7224

11. Solve for x: cos(x) = .5

12. Solve for : tan = 3

3

13. Solve for x: tan(x) = 5

14. Solve for : cos = 2

2

15. Solve for y: sin(y)=.8888

1 1

( ) sin( )

sin( )

inverse : sin( )

to solve for y:

sin sin

f x x

y x

x y

x

sin 1

1 1

( )

sin ( )

( ) sin ( )

y

y x

f x x

~ 3 ~

16. f() = cos(). For what

value of does f() =4

5?

17. f(B) = sin(B).

For what value of B does

f(B) =3

2?

18. f() = tan(). For what

value of does f() =.5?

Finding Trigonometric Function Values from Triangles

1. In right triangle ABC, C is the right angle.

If AB=13 and BC=12, find sinA and sinB.

2. In right triangle DEF, E is the right angle. If

DE=1 and DF=3, find tanF.

3. In right triangle XYZ, Z is the right angle. If

XY=10, YZ=5 3 , and XZ=5, find the following

trigonometric values.

SinX = SinY =

CosX = CosY =

TanX = TanY=

A

B C

E

F

D

X

Y Z

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Finding the values of the 3 trig ratios

Ex) If 4

sin5

x , find the values of the other two trigonometric functions.

Steps

1. Draw a right triangle and label the sides with the ratio

you are given. (Remember, the size of the triangle does not matter.)

2. Find the third side using Pythagorean theorem.

3. Find the other basic trig ratios using SOH-CAH-TOA.

1) If t is an acute angle and 6

tan7

t , find the values of the other two trigonometric

functions. Express your answers in simplest radical form.

2) If is an acute angle and 3

sin2

, find the value of tanθ. Express your answer in

simplest radical form.


cos13


functions.

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4) If is an acute angle and 15

cos4

, find sinθ. Express your answer in simplest radical

form.

5) If is an acute angle and tan 1 , find the values of the other two trigonometric



sin25


functions.


cos5



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Lesson #43 – Trigonometric Reciprocals and Quotients A2.A.55 Express and apply the six trigonometric functions as ratios of the sides of a right

triangle

A2.A.58 Know and apply the co-function and reciprocal relationships between trigonometric

ratios

Current trigonometric functions:

sin=

cos= tan=

What other ways could be make ratios out of the sides of a right triangle?

These other three ratios are also trigonometric functions. They are called the reciprocal

trigonometric functions because they are reciprocals of the three original trigonometric

functions. Each of the reciprocal functions also has its own name.

Life is not perfect. Sine and cosecant are

reciprocals and cosine and secant are reciprocals.

In this case, opposites attract.

5

1312

B

C

A

Name Abbreviation Ratio with sides

of a right

triangle

Relation to sine,

cosine, or

tangent

Value with

respect to <A

Cosecant

Secant

Cotangent

2029

21

B

CA

sinA= sinB=

cosA= cosB=

tanA= tanB=

cscA= cscB=

secA= secB=

cotA= cotB=

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E

F

D

A

B C

1. In right triangle ABC, C is the right angle.

If AB=25 and AC=7, find secA and secB.

2. In right triangle DEF, E is the right angle. If

DE=2 and DF=4, find cotF.

Finding Reciprocal Values from a Triangle

Finding the values of all 6 trig ratios

Ex) If 4

sin5

x , find the values of the other five trig ratios.

1. Draw a right triangle and label the sides

with the ratio you are given.

2. Find the third side using Pythagorean theorem.

3. Find the other basic trig ratios.

(You can find secant, cosecant, and cotangent quicker by

__________________________).

a. If t is an acute angle and 5

cos13

t , find the values of the other five trigonometric

functions.

We learned that

csc

Therefore, if 4

sin5

x ,

csc x

If we know a basic trig

ratio, we can flip that

value to find the

reciprocal ratio.

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b. If is an acute angle and 17

c15

cs , find the values of the other five trigonometric

functions.

c. If A is an acute angle and sec 3A , find the value of tan A in simplest radical form.

d. If β is an acute angle and 3

cot3

, find the value of sin(β) in simplest radical form.

Evaluating Reciprocal Trig Functions

To find the values of the reciprocal functions, you must enter it in reciprocal form in the

calculator. For example, to the nearest hundredth, 1

sec89 57.29cos89

.

1) Find sec45 to the nearest hundredth.

2) Find cot 45 to the nearest thousandth.

3) Find the value of csc65 to the nearest tenth.

4) f(x)=cot(x). Find the value of f(10°) to the nearest tenth.

5) f(x)=sec(x). Find the value of f(52°) to the nearest tenth.

Common Mistakes

1. Flipping the angle.

Ex) 1

csc32sin 32

,

not 1

sin32

.

2. Confusing the reciprocal

with inverse.

Ex) 1

csc32sin 32

,

not 1sin 32 .

3. Forgetting to check

your mode.

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The Quotient Relationships

Q: Why are there six trigonometric functions?

A: There are six different ratios you can make between the sides of a right triangle.

Even though there are six trigonometric functions, we can actually express them in terms of

sine and cosine ONLY. We already know that 1

cscsin

and 1

seccos

but how can we

express tanθ and cotθ in terms of sine and cosine?

Prove:

sintan

cos

Prove:

coscot

sin

sin

tancos

O

O H OHA H A A

H

coscot

sin

A

A H AHO H O O

H

These are the quotient identities. They will come in handy throughout the rest of the unit.

Let’s verify these quotient relationships with a couple of examples.

1) sin(46 )

cos(46 )

tan(46)°=

2) cos(46 )

sin(46 )

cot(46)°=

Use the quotient identities to complete the following problem.

3) If sinθ=4

5 and cosθ=

3

5, find tanθ and cotθ without drawing a triangle.

Quotient Relationships

sin

tancos

cos

cotsin

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Lesson #44 - Special Triangles and Trig Values

A2.A.59 Use the reciprocal and co-function relationships to find the value of the secant,

cosecant, and cotangent of 0°, 30°, 45°, 60°, 90°, 180°, and 270°

A2.A.56 Know the exact and approximate values of the sine, cosine, and tangent of 0°, 30°,

45°, 60°, 90°, 180°, and 270°

A2.A.58 Know and apply the co-function and reciprocal relationships between trigonometric

ratios

The two special triangles you learned last year were 30-60-90 and 45-45-90 triangles. They

are called “special” because they have specific ratios between their sides. Since all 45-45-90

triangles are similar and all 30-60-90 triangles are similar, size does not matter. The ratios are

always true.

Special Angle Trigonometric Values Summary Chart

Do your calculations for the exact values first on the

next page. Copy your final answers in into the table on

the left. You must know the bolded part of the table by

memory. The estimated values can be found using your

calculator. Round to the nearest thousandth when necessary.

Exact Values

30 45 60

sin

cos

tan

csc

sec

cot

Estimated Values

30 45 60

sin

cos

tan

csc

sec

cot

x

2x

90 60

30

x

x

4590

45

2x 3x

~ 11 ~

Use the two triangles to find the exact (in simplest radical form when necessary) trigonometric

values for sine and cosine. Use the reciprocal and quotient relationships to find the values for

tangent, secant, cosecant, and cotangent. All answers should be in simplest radical form. 30 45 60 Pattern/Method

sin

cos

tan

csc

sec

cot

x

2x

90 60

30

x

x

4590

45

2x

3x

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HINTS FOR MEMORIZATION: The 1-2-3, 3-2-1 Method Look at the table on page 10. What pattern do you notice in the sine values?

What pattern do you notice in the cosine values?

This pattern makes it much easier to memorize the

trigonometric values for the special angles. If you

memorize 1

sin 302

or even type sin(30°) into

your calculator to find out that it equals 1

2, you

can count from there to find the other values.

In the table to the right, find the sine and cosine

values using this method.

If you can use what you learned last lesson, you just need to memorize the sine AND

cosine trigonometric values for the special angles. All of the others you can find using

the quotient or reciprocal relationships. (It is still helpful to memorize the tangent

values.)

Quotient Relationships: tan = cot =

Go back to the table above and find tangent using the quotient relationship.

Reciprocal Relationships: csc = sec = cot =

Arithmetic with Special Angle Trigonometric Values

(Find the trig value before performing the operation.)

a) sin(30°)+cos(45°)=

b) =

c) sin(30°)(cos(30°)+cos(45))=

d) sin(45°)+tan(30)°=

e) cot(30°)tan(30°)=

f) csc(60°)+tan(45°)+sec(30°)=

30 45 60

sin

cos

tan

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Lesson #45 - Angle Measurements in Standard Position A2.A.57 Sketch and use the reference angle for angles in standard position

Standard Position for an angle: An angle with its initial (starting) side on the

positive x-axis and its vertex on the origin. The other side of the angle

is called the terminal (ending) side.

**Positive angles go __________________ while negative angles go

_______________.

How many degrees are in one rotation?

Why is that number helpful?

Draw angles in standard position with the following measures.

120 degrees

230 degrees

-40 degrees

80 degrees

-200 degrees -90 degrees

530 degrees

-400 degrees 360 degrees

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Quadrantal Angle: An angle with its terminal side on the axes.

The measurements of the first five positive

quadrantal angles would be: _______________

Coterminal angles: Angles with the same terminal (ending) side

Look at the previous page. What angle would be coterminal

with -200°?

Method for finding the smallest positive coterminal angle:

Add or subtract 360° until the angle is between 0 and 360.

We will need to find the smallest positive coterminal angle for the next section of this

lesson as well as the rest of the unit.

Ex) What is the smallest positive angle that is coterminal with an angle of 450°?

450°-360°=90°

1. What is the smallest positive angle that is coterminal with an angle of 700°?

2. What is the smallest positive angle that is coterminal with an angle of -460°?

3. What is the smallest positive angle that is coterminal with an angle of 120°?

4. Challenge: What is the smallest positive angle that is coterminal with an angle of 10,427

degrees?

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Reference Angle: an acute angle formed by the terminal side of the original angle and the x-

axis. All angles in standard position have reference angles EXCEPT the quadrantal angles.

Steps Ex) Find the reference angle for 500°. 1) If necessary, find the smallest

positive coterminal angle for the

given angle.

2) Draw the angle and decide in

which quadrant the terminal side

lies.

3) Find the angle measure between

the terminal side and the x-axis.

Quadrant I:

Quadrant II or III:

Quadrant IV:

Find the reference angle for each angle below. Draw a sketch to show your work. a) 285°

Quadrant IV

360-285 = 75°

b) 78° c) 270°

d) -17°

e) 268° f) 91°

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g) 243°

h) 306° i) -299°

j) -474°

k) 652° l) 1000°

m) 100°

n) -100° o) 540°

p) 552°

q) 78° r) -6000°

Expanding Our Understanding of Trigonometry

The other two angles in a right triangle must be in the following interval: 0 90

Sine and Cosine are functions. Are the domains sine and cosine 0 90 only?

Find sin(300°) and cos(-1000°).

How can what we know about right triangles and what we just learned about angles in standard

position make it possible to make the domains of sine and cosine all real numbers?

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Lesson #46 - Trigonometry on the Coordinate Plane A2.A.62 Find the value of the trigonometric functions, if given a point on the terminal side of

angle q (and much more)

The terminal side of an angle in standard position passes through the point, (3,4). Find the sine

of the angle. Then find the values of sinθ, cosθ, and tanθ.

When the angle is in standard position, the opposite side of the triangle is always the __

coordinate and the adjacent side of the triangle is always the __ coordinate.

In general, sin=_____, cos=_____, and tan_____.

In Lesson #42 as well as the end of the last lesson, you saw that there are sine, cosine, and

tangent values of angles larger than 90° and smaller than 0°.

Reference angles make it possible. From a reference angle, we can draw a reference triangle.

In the picture to the right, each triangle is a reference

triangle for an angle in one of the four quadrants. Notice,

each triangle has its base on the X-AXIS, not the y-axis.

A bow tie can help you remember this fact.

A reference triangle is simply the triangle that contains

the reference angle for the given angle.

Look at the general ratios with x and y you filled in above.

The signs of x and y are not always positive in the different quadrants. In the picture to the

right decide if the x and y values are positive or negative in each quadrant.

~ 18 ~

Draw an angle in standard position in each quadrant. Draw a reference triangle in each quadrant

to decide what trigonometric functions are positive or negative in that quadrant.

Just do sine, cosine, and tangent first.

The hypotenuse of the triangle is always positive.

- Go back and decide which of the reciprocal functions are positive or negative.

Here is a chart to help you remember this

pattern. There is a mnemonic statement that may

be helpful for remembering the positive trig

values (and their reciprocals) in each quadrant.

A S T C - All Students Take Calculus!

(We start the brainwashing early!)

Quadrant I

sin=

cos=

tan=

Quadrant III

sin=

cos=

tan=

Quadrant IV

sin=

cos=

tan=

sec=

csc=

cot=

sec=

csc=

cot=

sec=

csc=

cot=

sec=

csc=

cot=

Quadrant II

sin=

cos=

tan=

~ 19 ~

Working from a point on the terminal side of

1) The terminal side of an angle in standard position, θ, passes

through the point, (-4,3). Find the sin() and tan().

1. Plot the point and draw the angle in standard position.

2. Draw the reference triangle for that angle.

3. Label the side lengths based on the given point.

4. Use the Pythagorean theorem to find the hypotenuse.

5. Set up the trigonometric ratio you are asked to find. Be

sure to write the answer in simplest radical form if

necessary.

6. CHECK YOUR SIGNS!

3sin

5

3tan

4

2) The terminal side of , an angle in standard position, passes

through the point, 1, 5 . Find the values of the six

trigonometric functions.

3) The terminal side of an angle in standard position passes

through the point, (-9,-40). Find the cosecant of the angle.

Relating trigonometric functions of larger angles to their reference angles Steps

Draw the angle and label the quadrant.

Find the reference angle for the angle.

o (If necessary, use coterminal angles first.)

Draw a reference triangle.

The trigonometric values for the given angle will be the SAME as the reference

angle EXCEPT the sign might be different. Use the quadrant to decide.

3

-4

2 2 2

2

4 3

25

5

h

h

h

5

~ 20 ~

135°

sin(135°) is the same as sin(45°).

cos(135°) is the same as -cos(45°).

tan(135°) is the same as –tan(45°).

Check these answers on your calculator.

260°.

sin(260°) is the same as ________.

cos(260°) is the same as ________.

tan(260°) is the same as ________.

Check these answers on your calculator.

The directions to this type of question can be the most confusing part. It will often say,

“Write the given function as a function of an acute angle.”

You are really just doing what we did in the above chart. You will even get to the point where

you do not need to draw the axes anymore.

Write the given function as a function of an acute angle.

1) sin132

2) cos280

3) cos218

4) tan( 20 )

5) tan 225

6) sin( 125 )

The steps for every problem can be summarized with some combination of the

following steps

1) Draw a picture.

2) Use coterminal angles until the angle is smaller than 360°.

3) Find the reference angle.

4) Set up the trigonometric ratio(s).

5) Decide if the trigonometric function(s) are positive or negative in that

quadrant.

+

-

+

The angle is in

the 2nd

quadrant so

sine will be

positive, but

cosine and

tangent will be

negative.

~ 21 ~

Lesson #47 - Radian Measure A2.M.1 Define radian measure

A2.A.61 Determine the length of an arc of a circle, given its radius and the measure of its

central angle

A2.M.2 Convert between radian and degree measures

Why do we use degees to measure angles?

Are there other ways to measure angles?

Image Angle Measurement Number in a

full rotation

Degrees

“Clockians”

“Petaloids”

Radians

~ 22 ~

What is another way that you know 2π radii (plural for radius) fit perfectly around the outside

of a circle?

Arc length: (symbol: s) The length of an arc which is a section or part of a circle's perimeter

(a.k.a. circumference)

Central Angle: An angle positioned with its vertex at the center of a

circle and its endpoints on the circle

Subtend: to intersect or be opposite

One Radian: (abbreviation: rad) The measure of a central angle that

subtends an arc length that is one radius long.

http://id.mind.net/~zona/mmts/trigonometryRealms/radianDemo1/RadianDemo1.html

How many radii fit around a circle?

THE MEASURE OF THE CENTRAL ANGLE IS ONE RADIAN

WHEN THE ARCLENGTH IS EQUAL TO THE RADIUS.

The size of the circle

does not affect the

number of radians in the

circle because the radius

and the circumference

will grow or shrink in

proportion to each other.

http://id.mind.net/~zona/mmts/trigonometryRealms/radianDemo1/RadianDemo1.html

~ 23 ~

New Formula s

r

= central

angle in

radians

s = arc length

r = radius

Note: From now on, if an angle does not have a degree symbol, you must

assume it is measured in radians.

An angle is 1 radian when the _________ is equal to the ________.

If the arclength (s) is 5 inches and the radius (r) is 5 inches the radian measure of the angle

would be ________.

In other words, the radian measure of an angle is equal to the ratio of the arclength (s) to the

radius (r).

This gives us the formula, rads

r .

If we use as the variable for the angle, this gives us the formula, s

r .

NOTE: must be in radians to use this formula, not degrees.

Draw the following conditions: If the radius of the circle to the right

is 5 and the arc length subtended by the central angle is 10, would

be ___ radians. This makes sense because ___ radii fit around the

arclength.

1. Find the arclength if radius is 6 inches and the measure of the central angle is 3 radians.

2. Find the exact value of the central angle if the radius of the circle is

10 inches and the arclength is 5π inches.

3. Find the radius of a circle if the arclength is 12 in and the central angle is 5

6

to the nearest

tenth.

4. Find the arclength of a circle if the radius is 2 cm and the measure of the central angle is

3.2 radians.

s = arclength

ø = angle in radians

r = radius

~ 24 ~

Converting between radian and degree measure A full rotation has ___ degrees or ___ radians.

Therefore the ratio of degrees to radians is ____:______. Divide both sides by 2.

____:______.

We will use this fact to convert between radians and degrees.

Converting from degrees to radians

Multiply by 180

Ex) Convert 120° to radians.

Exact answer: 2

120180 3

Nearest Hundredth: 120 2.09180

Converting from radians to degrees

Multiply by 180

Ex) Convert 5

12

radians to degrees.

5 180

7512

1) Convert to radians. Write as an exact answer and rounded to the nearest tenth.

a. 30º b. 260º

c. 45º d. 180º

2) Convert each radian measure to degrees. Round to the nearest degree when necessary.

a.

3

b. 3

5

c. 2.45

d. 7

5

You can enter 120/180 in your

calculator, change it to a

fraction, and put π after it.

To

radians:

“put in π

take out

180°

To °:

“put in

180° take

out π”

~ 25 ~

Lesson #48 – The Unit Circle A2.A.60 Sketch the unit circle and represent angles in standard position

A2.A.59 Use the reciprocal and co-function relationships to find the value of the secant,

cosecant, and cotangent of 0°, 30°, 45°, 60°, 90°, 180°, and 270°

A2.A.56 Know the exact and approximate values of the sine, cosine, and tangent of 0°, 30°,

45°, 60°, 90°, 180°, and 270°

You learned in Lesson #42 that the size of the triangle does not matter when finding the

trigonometric functions of an angle.

You learned in Lesson #46 that when given a point on an angle in standard postion siny

h and

cos .x

h Draw a picture of this concept in the axes provided.

Since the size of the triangle does not matter, how could

we make cos =x and sin =y?

A way to make sure our triangle always has a hypotenuse of 1 is to contain the triangle in a

circle with a radius of 1, centered at (0,0). This is the parent relation for circles we learned about in unit 4.

Its equation is 2 2 1x y and we call it the unit circle because it

has a radius of one unit.

On the axes draw an angle with its terminal side in the

first quadrant.

Draw the unit circle.

Draw a reference triangle using the point

where the angle intersects the circle

as the end of your hypotenuse.

Label the sides of the triangle.

sin cos

This is only true

when the point is

on the unit circle!

~ 26 ~

4/51

3/5

How is the unit circle helpful?

a. It allows us to work with trigonometric functions without using a triangle because the

hypotenuse is always one. All we need is a point on the unit circle, (x,y).

b. The x-value of the point on the unit circle is the same as the cosine of the angle.

c. The y-value of the point on the unit circle is the same as the sine of the angle.

d. It allows us to find the trigonometric values of the quadrantal angles (90°, 180°, etc.).

e. It is the basis for creating the function graphs of sine and cosine (next lesson).

f. It is the easiest way to find many trigonometric identities that we will learn about at the

end of the year.

Information from a Point on the Unit Circle If you know a point on the unit circle intersected by , you know sin and cos.

1) The terminal side of intersects the unit circle at the

point 3 4

,5 5

. What is Sin ? What is Cos ?

2) The terminal side of intersects the unit circle at the point

2 21,

5 5


~ 27 ~


point 1 15

,4 4



point 3 1

,2 2


You can also find the values of other trigonometric functions

from the point on the unit circle.

5) The terminal side of intersects the unit circle at the point 9 40

,41 41

. What is Tan ?


point 5 12

,13 13

. What is sec ?

~ 28 ~


point 5 2

,3 3

. What is csc ?


approximate point, 8 15

,17 17

. What is cot ?

Quadrantal Angles Draw a unit circle on the axes below to find the trigonometric values for the quadrantal angles.

This is really the only way to conceptualize sin90°, cos180°, etc. because you cannot draw a

reference triangle for a quadrantal angle.

Point

on Unit

Circle

sin cos tan sec csc cot

0°,

360°

(1,0)

0 1 sin 00

cos 1

1 11

cos 1

1 1.

sin 0undef

cos 1.

sin 0undef

90°

180°

270°

To find the values for

the other four

trigonometric functions

use the reciprocal and

quotient relationships!

All of the values

in the chart can

be checked on the

calculator.

~ 29 ~

Use the unit circle to find the following trigonometric values. Show your work.

9) sin90

10) cos270

11) tan180

12) sec180

13) csc0

14) cot180

15) What point on the unit circle is intersected by an angle of 0°?


Exact Values and Special Angles If is a special angle, we do not need to round the coordinates of the point on the unit

circle because we know the exact value of sin and cos. Remember to first find the

reference angle and consider the quadrant for the sign of the answer.




~ 30 ~

The Unit Circle Diagram

The unit circle diagram

helps you start to think

about special angles in the

other quadrants as well as

the radian measures of the

special angles.

Patterns:

1) What do you notice

about the points on the

angles with a reference

angle of 30°?


about the points on the

angles with a reference

angle of 45°?


about the points on the angles with a reference angle of 60°?

4) What do you notice about the radian measure of the angles with a reference angle of 30°?



~ 31 ~

FOR EACH OF THE FOLLOWING PROBLEMS, DRAW A SKETCH OF THE UNIT CIRCLE.

Let’s look at how the unit circle diagram summarizes information about the special angles.

We will use 210° as an example.

*We could convert the angle to radians: 7

210180 6

*We could find the sine and cosine values of 210° using reference angles and the special

angle ratios.

1sin(210 ) sin(30 )

2

3cos(210 ) cos(30 )

2

All of this information is displayed concisely on the unit circle diagram.



22) What point on the unit circle is intersected by an angle of -120°?



You can also start with a point on the unit circle intersected by a special angle or a quadrantal

angle and work backwards to find the angle measure in either degrees or radians.

25) Find the measure of the angle in both degrees and radians that intersects the

point 1 3

,2 2

on the unit circle?

The angle is in the fourth quadrant (+,-).

1

cos2

when θ=60°. (3

sin2

when θ=60° also).

Therefore θ=360-60=300°

In radians: 5

300180 3

~ 32 ~


point (0,-1) on the unit circle.

27) Find the measure of the angle that intersects the point 2 2

,2 2

on the unit

circle.


point 3 1

,2 2

on the unit circle.


point (1,0) on the unit circle.


point 3 1

,2 2

on the unit circle.

31) Find the measure of the angle in both degrees and radians that intersects the point

1 3,

2 2

on the unit circle.

~ 33 ~

~ 34 ~

Lesson #49 – Graphing Sine and Cosine Since sin() and cos() are functions, they can be graphed. Their input is

the angle measure and their output is the sine or cosine value depending on

the graph. The unit circle diagram is especially helpful in generating these

graphs.

Trigonometric graphs are usually drawn with radians for the angle measure.

You will want to memorize the degree/radian conversions that are already

recorded in the table to the right. Continue the pattern for the other

degree measures.

Degrees Radians

-450

-360

-270

-180

-90

0 0

90 2

180

270 3

2

360 2

450

540

630

720

810

~ 35 ~

Using the chart, create a graph for y=sin on the interval 0 2 . Use the unit circle to

find the sine values for the quadrantal angles. Extend the pattern so that 3

a) What is the domain of y=sinθ?

b) What is the range of y=sinθ?

c) Is sinθ a 1-to-1 function? Justify your answer.

d) What is the y-intercept of y=sinθ?

e) What is the maximum value of y=sinθ?

f) What is the minimum value of y=sinθ?

g) The distance from the x-axis (midline) to the maximum value of sinθ is known as the

amplitude.

What is the amplitude of y=sinθ?

h) The period of a trig. function is the interval length of one complete cycle of the graph.

What is the period of y=sinθ?

Input Output

Sin

0

2

3

2

2

Suggested Scales when

graphing y=sin(x) and y=cos(x)

x-axis: 2 spaces is 2

(90°).

y-axis: 2 spaces is 1

~ 36 ~

Graphing trigonometric functions on the calculator

1. Make sure your calculator is in RADIAN mode.

2. Type the function in Y1.

3. Press Zoom and choose #7, Zoomtrig.

4. If you press window

5. Adjust the X-min and X-max to fit the specified interval such as 0 2

Window:

Xmin: -6.15 . . .

Xmax: 6.15 . . .

XScl: 1.57. . .

Ymin: -4

Ymax: 4

Y-scl: 1

Window in terms of π:

Xmin: about -2π

Xmax: about 2π

XScl: exactly 2

Ymin: -4

Ymax: 4

Y-scl: 1

~ 37 ~

Using the chart, create a graph for y=cos on the interval 0 2 . Use the unit circle to find

the cosine values for the quadrantal angles. Extend the pattern so that 3

a) What is the domain of y=cosθ?

b) What is the range of y=cosθ?

c) Is y=cosθ a 1-to-1 function? Justify your answer.

d) What is the y-intercept of y=cosθ?

e) What is the maximum value of y=cosθ?

f) What is the minimum value of y=cosθ?

g) The distance from the x-axis to the maximum value of y=cosθ is known as the amplitude.

What is the amplitude of y=cosθ?

i) The period of a trig. function is the interval length of one complete cycle of the graph.

What is the period of y=cosθ?

Summary for checking trig graphs on your calculator 1. Make sure you are in radian mode unless the problem indicates degree mode.

2. Type in Y=

3. Press ZoomTrig

4. Adjust the Window manually if the interval is not 2 2 .

5. Graph the function. The x-scale should go by 2

(usually for every 2 spaces) and the y-

scale should go by 1 to match your calculator.

Input Output

Cos

0

2

3

2

2

~ 38 ~

Lesson #50 – Graphs of Tangent and the Reciprocals A2.A.71 Sketch and recognize the graphs of the functions y=sec x, y = csc x, y = tan x,

y = cot x

Note: The four functions we are working with today can be written as rational expressions with

sine, cosine or both because they are the quotient and reciprocal functions.

What are the restricted values for1

( )f xx

?

What are the restricted values for 1

( )2

g xx

?

Vertical asymptotes are vertical lines which correspond to the zeroes of the

denominator of a rational function. The graph of the function gets closer and closer to

these lines, but it will never touch or cross the line.

Draw a sketch of g(x) using your calculator. Draw a vertical asymptote at x = -2.

What are the restricted values for1

( ) csc( )sin( )

f x xx

? (You can use the unit circle to

help you answer this question).

Note: If you have a TI-83 calculator, it incorrectly tries to connect the infinity to the

negative infinity which creates a vertical line that is not really part of the graph. You can

see this with the graph of 1

( )2

f xx

. TI-84 calculators do not make this error.

Steps:

1. Use the unit circle to find the trig. values for the quadrantal angles. Use a table.

2. Draw the vertical asymptotes where the function is undefined.

3. Plot the other points.

4. Use your calculator as a guide to sketch the rest of the graph. Make sure you are in

radian mode and use #7: ZoomTrig.

http://www.purplemath.com/modules/rtnldefs.htm

~ 39 ~

Graph for y=csc. Use the unit circle to find the cosecant values for the quadrantal angles.

Extend the pattern so that 2 2

a) What is the domain of y=cscθ?

b) What is the range of y=cscθ?

Graph y=sec. Use the unit circle to find the secant values for the quadrantal angles. Extend

the pattern so that 2 2

a) What is the domain of y=secθ?

b) What is the range of y=secθ?

Input Output

csc

Input Output

sec

~ 40 ~

Graph for y=tan. Use the unit circle to find the tangent values for the quadrantal angles.


c) What is the domain of y=tanθ?

d) What is the range of y=tanθ?

Graph y=cot. Use the unit circle to find the cotangent values for the quadrantal angles.


c) What is the domain of y=cotθ?

d) What is the range of y=cotθ?

Input Output

tan

Input Output

cot

~ 41 ~

Lesson #51 – Transformations on Sine and Cosine A2.A.69 Determine amplitude, period, frequency, and phase shift, given the graph or

equation of a periodic function

A2.A.70 Sketch and recognize one cycle of a function of the form y =A sinBx or y=AcosBx

A2.A.72 Write the trigonometric function that is represented by a given periodic graph

Fill in the table below. 1-5 are review from last unit. 6 is a new transformation.

Function Notation Transformation 1. -f(x)

2. f(-x)

3. af(x) 0 < a < 1:

a > 1:

4. f(x+a) +a:

-a:

5. f(x)+a +a:

-a:

6. f(ax) 0 < a < 1:

a > 1:

Why do horizontal changes (4 and 6) work the opposite of what you would think? Do the

examples below to find out. When answering the questions, think of a value for a such as 2.

x=5

If I want the value of x to be increased by a, show

would you would have to do to x

y=5

If I want the value of y to be increased by a, show

would you would have to do to y. Solve for y.

x=5

If you want the value of x to be multiplied by a, show

would you would have to do to x.

y=5

If you want the value of x to be multiplied by a, show

would you would have to do to y. Solve for y.

The above chart is showing that you need to perform the inverse operation on the variable so

that the desired change happens when you solve for the variable.

With functions, we solve for y. Therefore the change ends up being the operation you would

expect. We do not solve for x, so you still see the inverse operation with x.

~ 42 ~

The transformations we will be doing on sine and cosine are bolded and underlined in the table

on the previous page. For each of the following equations identify the transformation(s) on

that have occurred on either sine or cosine.

Note: When working with trigonometric functions, horizontal shifts (left or right) are referred

to as phase shifts when the equation is in the form y=asin(b(x+c)) or y=acos(b(x+c)). Use this

term when completing the section below.

1. 5sin3y x

2. 1

2cos2

y x

3. 3siny x

4. cos 2y x

5. 100cos 32

y x

6. 2

cos73

y x

7. sin6

y x

8. = cos 54

xy

Tricky!

~ 43 ~

Amplitude, Period, and Frequency Amplitude: The height of the a trigonometric function from the midline to its maximum y-value.

The amplitude is equal to the vertical stretch/compression.

Period: The length of one cycle

The period is equal to divided by the horizontal compression/stretch.

Frequency: The number of cycles (periods) from 0 to .

The frequency is equal to the horizontal compression/stretch.

Example: y = 5cos(3x)

The amplitude is 5. The parent cosine curve has an amplitude of 1, so a vertical stretch of 5

would result in an amplitude of 5.

The period is 2

3

. The parent cosine curve has a period of 2 . Since this graph is

horizontally compressed by 3, the period will become 3 times shorter.

The frequency of this graph is 3. The parent cosine curve has a frequency of 1. Since this

graph is horizontally compressed by 3, there will be 3 cosine curves over the interval from 0 to

2 .

Go back to the previous page and identify the amplitude, period, and frequency of each of the

functions.

y = Asin(B(x+C)) or y = Acos(B(x+C))

Amplitude = |A|

Period = 2

B

Frequency = B

Phase Shift = C

~ 44 ~

Graphing Transformations on Sine and Cosine Graph one period of y=sinx and y=cosx. Use the unit circle to make your table. These tables

and graphs are the starting point for every transformation on sine or cosine. If you do not

know how to graph them, the rest of this lesson will be very difficult.

y=sinx y=cosx

For this course, you only have to graph one period of transformations of sine and cosine in the

form y=AsinBX or y=AcosBx.

Steps

1. Start with one of the basic tables for either sine or cosine.

2. A will result in a vertical stretch/compression. Multiply the y-values by A.

3. B will result in a horizontal compression/stretch. Since a horizontal compression/stretch

works in the opposite from the way you would expect, you need to divide the x-values by

B. 4. A negative in front of the equation means the function was reflected over the x-axis.

Therefore negate the y-values.

Scales

For trig. graphs, we typically label the x-axis with 2

for every 2 spaces.. For transformations,

it is easiest to label the x-axis with 2

B

for every 2 spaces instead.

Adjust the interval on the y-axis based on the value of A if necessary.

x y

x y

~ 45 ~

5cos3y x

Horiz. comp. of 3. (Divide x-values by 3).

Vertical stretch of 5. (multiply y-values by 5).

x cosx

0 1

2

0

-1

3

2

0

2 1

x 5cos3x

12sin

2y x

~ 46 ~

3cosy x

( ) sin6

f x x

~ 47 ~

Graph one period of a sine function with an

amplitude of 4 and a period of π.

Graph one period of a cosine function with an

amplitude of

1

2 and a period of 6π.

Note: Many of the transformations drawn above look similar to each other, but if they were

graphed with the same scale they would look very different.

You can set your calculator with the minimums, maximums, and scales you find in the table to

check your graph. Make sure you are in radian mode.

~ 48 ~

Writing the equation of a trigonometric function from a graph

Sine vs Cosine and reflection in x-axis:

The y-intercept of a sine curve is __. From that point the sine curve first (circle one)

increases/decreases (circle one).

If the sine curve has been reflected over the x-axis, the sine curve first increases/decreases

(circle one).

The y-intercept of a cosine curve is ________________________.

Therefore, the y-intercept of a cosine curve that has been reflected over the x-axis is

_________________________.

Finding A: The distance from the midline (usually y=0) and the maximum value, or the amplitude,

of the graph is the value of A.

Finding B: Determine the period of the function from the graph. Use the formula, 2

periodB

and solve for B.

An alternative method that works for some graphs is to think about how “fast” the graph is

going or its frequency which is equal to B. The parent and cosine functions have a period of 2 .

If a graph has a period of , it is going double speed so B=2. (Horizontally compressed by 2).

If a graph has a period of 4 , it is going half speed so B=1

2. (Horizontally stretched by

1

2).

You can also identify these qualities from the graph of the trigonometric function. Only

complete the 2nd and 3rd columns to start.

Graph Identify the

amplitude and

period, and B

(frequency)

Write an

equation for

the graph

Challenge: Write

an equation for

the other trig.

function using a

phase shift

Amplitude =A= 2

Period: 2π

22

1

B

B

(Normal speed: freq.

= 1)

Sine or Cosine

based on the

y-intercept?

Sine

Refection in x-

axis? Yes

Y=-2sin(1x)

Same equation with

cosine:

shift Y= -2cos(x)

2

to the right.

2cos2

y x

~ 49 ~

Graph each of the equations you find

on the calculator to see if they match

the given graph. Set your window and

scales to match the given graph.

The fourth column is an advanced skill that will

probably not be on the regents.

Documents

Lesson #42 Three Basic Trigonometric Functions€¦ · trigonometric functions because they are reciprocals of the three original trigonometric ... Sine and cosecant are reciprocals