Trigonometric Functions: The Unit Circle

Trigonometric Functions: The Unit Circle

Section 4.2

Objectives• Identify a unit circle and describe its

relationship to real numbers.• Evaluate trigonometric functions using the

unit circle.• Recognize the domain and range of sine and

cosine functions• Find the exact values of the trig functions at

/4• Use even and odd trig functions• Recognize and use fundamental identities• Periodic functions

Trigonometric Ratios• The word trigonometry originates from

two Greek terms, trigon, which means triangle, and metron, which means measure. Thus, the study of trigonometry is the study of triangle measurements.

• A ratio of the lengths of the sides of a right triangle is called a trigonometric ratio. The three most common trigonometric ratios are sine, cosine, and tangent.

Trigonometric Ratios

Only Apply to Right Triangles

In right triangles : • The segment across from the right angle ( ) is labeled the hypotenuse

“Hyp.”.

• The “angle of perspective” determines how to label the sides.• Segment opposite from the Angle of Perspective( ) is labeled “Opp.”• Segment adjacent to (next to) the Angle of Perspective ( ) is labeled

“Adj.”.

* The angle of Perspective is never the right angle.

ACA

B C

Hyp.Angle of Perspective

Opp.

Adj.

ABBC

Labeling sides depends on the Angle of Perspective

A

A

B C

Angle of Perspective

Hyp.

Opp.

Adj.

If is the Angle of Perspective then ……

AC Hyp

BC Opp

AB Adj

*”Opp.” means segment opposite from Angle of Perspective

“Adj.” means segment adjacent from Angle of Perspective

If the Angle of Perspective is

CA then

AC Hyp

BC Opp

AB Adj

A

B COpp

HypAdj

thenA

B C

Opp

Adj

Hyp

AC Hyp

AB Opp

BC Adj

The 3 Trigonometric Ratios

• The 3 ratios are Sine, Cosine and Tangent

Opposite SideSine RatioHypotenuse

sin Adjacent SideCo e RatioHypotenuse

Opposite SideTangent RatioAdjacent Side

Chief SohCahToa

The Amazing Legend of…

http://decaco.com/new_page_191.htm

Chief SohCahToa• Once upon a time there was a wise old

Native American Chief named Chief SohCahToa.

• He was named that due to an chance encounter with his coffee table in the middle of the night.

• He woke up hungry, got up and headed to the kitchen to get a snack.

• He did not turn on the light and in the darkness, stubbed his big toe on his coffee table….

Please share this story with Mr. Gustin for historical credibility.

Trigonometric RatiosTo help you remember these

trigonometric relationships, you can use the mnemonic device, SOH-CAH-TOA, where the first letter of each word of the trigonometric ratios is represented in the correct order.

A

C B

bc

a

Sin A = Opposite side SOH HypotenuseCos A = Adjacent side CAH HypotenuseTan A = Opposite side TOA Adjacent side

SohCahToa

hypotenuseoppositesin

hypotenuseadjacentcos

adjacentoppositetan

Soh

Cah

Toa

The six trigonometric functions of a right triangle, with an acute angle , are defined by ratios of two sides of the triangle. The sides of the right triangle are:

the side opposite the acute angle ,

the side adjacent to the acute angle ,

and the hypotenuse of the right triangle.

The trigonometric functions are

sine, cosine, tangent, cotangent, secant, and cosecant.

opp

adj

hyp

θ

sin 𝜃=𝑜𝑝𝑝h𝑦𝑝 cos𝜃=

𝑎𝑑𝑗h𝑦𝑝 tan𝜃=

𝑜𝑝𝑝𝑎𝑑𝑗

csc 𝜃=h𝑦𝑝𝑜𝑝𝑝 sec𝜃=

h𝑦𝑝𝑎𝑑𝑗 cot 𝜃=

𝑎𝑑𝑗𝑜𝑝𝑝

A unit circle is a circle of radius 1, with its center at the origin of a rectangular coordinate system.

The equation of this unit circle is

The length of the intercepted arc is t. This is also the radian measure of the central angle. Thus, in a unit circle, the radian measure of the central angle is equal to the length of the intercepted arc. Both are given by the same real number t.

The Unit Circle• Here we have a unit

circle on the coordinate plane, with its center at the origin, and a radius of 1.

• The point on the circle is in quadrant I.

The Unit Circle

• Connect the origin to the point, and from that point drop a perpendicular to the x-axis.

• This creates a right triangle with hypotenuse of 1.

The Unit Circle

sin( ) y y1

x

y1

is the angle of rotation

• The length of its legs are the x- and y-coordinates of the chosen point.

• Applying the definitions of the trigonometric ratios to this triangle gives

• The coordinates of the chosen point are the cosine and sine of the angle . – This provides a way to define functions

sin() and cos() for all real numbers .

– The other trigonometric functions can be defined from these.

The Unit Circle

sin( ) y y1

Trigonometric Functionssin( ) y

x

y1

is the angle of rotation

These functions are reciprocals of each other.

Around the Circle

• As that point moves around the unit circle into quadrants I, II, III, and IV, the new definitions of the trigonometric functions still hold.

The Unit CircleOne of the most useful tools in trigonometry

is the unit circle. It is a circle, with radius 1 unit, that is on the

x-y coordinate plane.

30º -60º -90º

The hypotenuse for each triangle is 1 unit.

45º -45º -90º

30º

60º1

45º

45º

1

The angles are measured from the positive x-axis (standard position) counterclockwise.In order to create the unit circle, we must use the special right triangles below:

cos

sin

The x-axis corresponds to the cosine function, and the y-axis corresponds to the sine function.

1

You first need to find the lengths of the other sides of each right triangle...

30º

60º1

45º

45º

1

32

22

22

12

Usefulness of Knowing Trigonometric Functions of Special Angles: 30o, 45o, 60o

• The trigonometric function values derived from knowing the side ratios of the 30-60-90 and 45-45-90 triangles are “exact” numbers, not decimal approximations as could be obtained from using a calculator

• You will often be asked to find exact trig function values for angles other than 30o, 45o and 60o angles that are somehow related to trig function values of these angles

Now, use the corresponding triangle to find the coordinates on the unit circle...

(1, 0)

sin

cos

(0, 1)

(–1, 0)

(0, –1)

32

12

30º

What are thecoordinates ofthis point?

(Use your30-60-90triangle)

This coorespondsto (cos 30,sin 30)

(cos 30, sin 30)

32

,12

Now, use the corresponding triangle to find the coordinates on the unit circle...

(1, 0)

sin

cos

(0, 1)

(–1, 0)

(0, –1)

(cos 30, sin 30)

32

,12



22

22

45º

22

,2

2

(cos45, sin 45)

You can use your special right triangles to find any of the points on the unit circle...

(1, 0)

sin

cos

(0, 1)

(–1, 0)

(0, –1)

(cos 30, sin 30)

32

,12

22

,2

2

(cos45, sin 45)



32

12 1

2,

32

(cos 270, sin 270)

Use this same technique to complete the unit circle.

(1, 0)

sin

cos

(0, 1)

(–1, 0)

(0, –1)

(cos 30, sin 30)

32

,12

22

,2

2

(cos45, sin 45)

12

,3

2

(cos 300, sin 300)

Unit Circle

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

0

2π

π

23π

Unit Circle

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

30°

6π

)21

,23

(6

5π)

21

,23

(

30°

30°

67π

)21

,23

( 6

11π

30°

)21

,23

(

Unit Circle

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

60°

3π

)23

,21

(32π

60°

60°

34π

35π

60°

)23

,21

(

)23

- ,21

(

)23

- ,21

(

Unit Circle

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

45°

4π )

22

,22

(4

3π

45°

45°

45π

47π

45°

)22

,22

(

)22

- ,22

(

)22

- ,22

(

Here is the unit circle divided into 8 pieces. Can you figure out how many degrees are in each division?

45°

We can label this all the way around with how many degrees an angle would be and the point on the unit circle that corresponds with the terminal side of the angle. We could then find any of the trig functions.

45°

22,

2290°

1,0

0°

135°

22,

22

180° 0,1

225°

270°315°

22,

22

22,

22

1,0

225sin22

0,1

These are easy to

memorize since they

all have the same value

with different

signs depending

on the quadrant.

Can you figure out what these angles would be in radians?

The circle is 2 all the way around so half way is . The upper half is divided into 4 pieces so each piece is /4.

45°

22,

2290°

1,0

0°

135°

22,

22

180° 0,1

225°

270°315°

22,

22

22,

22

1,0

4

7sin 22

0,14

2

43

45

23

47

Here is the unit circle divided into 12 pieces. Can you figure out how many degrees are in each division?

30°

We can again label the points on the circle and the sine is the y value, the cosine is the x value and the tangent is y over x.

30°

21,

23

90° 1,0

0°

120°

180° 0,1

210°

270°

330°

1,0

330cos23

0,1

You'll need to

memorize these too but you can see

the pattern.

60°150°

240°300°

23,

21

23,

21

23,

21

21,

23

21,

23

21,

23

23,

21240sin

23

Can you figure out what the angles would be in radians?

30°

It is still halfway around the circle and the upper half is divided into 6 pieces so each piece is /6.

30°

21,

23

90° 1,0

0°

120°

180° 0,1

210°

270°

330°

1,0

0,1

60°150°

240°300°

23,

21

23,

21

23,

21

21,

23

21,

23

21,

23

23,

21

We'll see

them all put

together on the

unit circle on the next screen.

6

You should memorize

this. This is a great

reference because you can

figure out the trig

functions of all these angles quickly.

23,

21

(1,0)

(0,1)

(0,-1)

(-1,0)

23,

21

sin

cos

We divide the unit circle into various pieces and learn the point values so we can then from memory find trig functions.

tan

23,

21

22

22

So if I want a trig function for whose terminal side contains a point on the unit circle, the y value is the sine, the x value is the cosine and the tangent is .

22,

22

1

22

22

(1,0)

(0,1)

(0,-1)

(-1,0)

We know all of the sides of this triangle. The bottom leg is just the x value of the point, the other leg is just the y value and the hypotenuse is always 1 because it is a radius of the unit circle.

21

231

sin

cos 21

121

Notice the sine is just the y value of the unit circle point and the cosine is just the x value.

tan 3

2123

23,

21

23

123

Finding Values of the Trigonometric Functions

Find the values of the six trig functions at

What are the coordinates? (0,1)

sin 𝜋2 =¿¿𝑦¿1 csc 𝜋2 =¿

1𝑦¿1

cos 𝜋2 =¿𝑥¿0 sec 𝜋2 =¿¿ 1𝑥¿𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑

tan 𝜋2 =¿¿𝑦𝑥¿𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 cot 𝜋2 =¿¿

𝑥𝑦¿0

Finding Values of the Trigonometric Functions

Find the values of the six trig functions at

Let’s think about the function

What is the domain? (domain means the “legal” things you can put in for ). You can put in anything you want

so the domain is all real numbers.

What is the range? (remember range means what you get out of the function). The range is: -1 sin 1

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

Let’s look at the unit circle to answer that. What is the lowest and highest value you’d ever get for sine? (sine is the y value so what is the lowest and highest y value?)

Let’s think about the function f() = cos

What is the domain? (domain means the “legal” things you can put in for ). You can put in anything you want

so the domain is all real numbers.

What is the range? (remember range means what you get out of the function). The range is: -1 cos 1

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

Let’s look at the unit circle to answer that. What is the lowest and highest value you’d ever get for cosine? (cosine is the x value so what is the lowest and highest x value?)

Even and Odd Trig FunctionsThe cosine and secant functions are even. Think “same as”…even…get it?

The sine, cosecant, tangent, and cotangent functions are odd. Think “opposite”.

Now let’s look at the unit circle to compare trig functions of positive vs. negative angles.

?3

cos isWhat

?3

cos isWhat

Remember negative angle means to go clockwise

21

21

23,

21

coscos Recall that if we put a negative in the function and get the original back it is an even function.

?3

sin isWhat

?3

sin isWhat

23

23

23,

21

sinsin Recall that if we put a negative in the function and get the negative of the function back it is an odd function.

?3

tanisWhat

?3

tanisWhat

23,

21

3

3

Using Even and Odd Functions to Find Values of Trig Functions

Find the value of:

Is cosine an even or odd function? It is even. It has the same value as , which is the x-coordinate for , and that is

Answer:

Let’s check with the calculators.

Using Even and Odd Functions to Find Values of Trig Functions

Find the value of:

Is tangent an even or odd function? It is odd. It has the opposite sign value as tan, which is .

Answer:

Let’s check with the calculators.

Reciprocal Identities

Reciprocal Identities

Quotient Identities

Using Quotient and Reciprocal Identities

Given and , find the value of each of the four remaining trig functions.

We need to find tangent, cotangent, secant, and cosecant.

Finish out the problem.

25

√215



Now we need to find cotangent.


√21525



Now we need to find secant.

sec




Now we need to find cosecant.

csc


Pythagorean IdentitiesThe equation of a unit circle is

Since and , then

Pythagorean Identities

Pythagorean Identities

Using a Pythagorean IdentityGiven that and , find the value of cos using a trig identity.

We can find the value of using the Pythagorean Identity.

cos𝜃=√ 1625=

45

Periodic FunctionsA periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π or radians. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena.

A period of is one revolution around the unit circle. A period of is one-half revolution.

Periodic Properties of the Sine and Cosine Functions

and

The sine and cosine functions are periodic functions and have a period of .

The secant and cosecant functions are also periodic functions and have a period of .

Periodic Properties of the Tangent and Cotangent Functions

and

The tangent and cotangent functions are periodic functions and have a period of .

23,

21

Sine and cosine are periodic with a period of 360 or 2.

We see that they repeat every so the tangent’s period is .

Let's label the unit

circle with values of

the tangent. (Remember this is just

y/x)

0

33

1

3

undef3

1

33

0

33

1

3 undef

33

1

3

Reciprocal functions have the same period.

PERIODIC PROPERTIESsin( + 2) = sin cosec( + 2) = cosec

cos( + 2) = cos sec( + 2) = sec tan( + ) = tan cot( + ) = cot

4

9tan This would have the same value as

4tan 1

(you can count around on unit circle or subtract the period twice.)

EXAMPLES: Evaluate the trigonometric function using its period as an aid

cos5

sin 94

cos 83

sin196

Using A Calculator to Evaluate Trigonometric Functions

Refer to pages 485 – 486

Go to modeSet to radians

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Trigonometric Functions: The Unit Circle