1. Trigonometric Functions: The Unit Circle
Section 5.2

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

3. 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.

4. Only Apply to Right Triangles
Trigonometric Ratios
Only Apply to Right Triangles

5. 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.
Hyp.
Opp.
Angle of Perspective
Adj.

6. Labeling sides depends on the Angle of PerspectiveIf
is the Angle of Perspective then ……
Angle of Perspective
Hyp.
Adj.
Opp.
*”Opp.” means segment opposite from Angle of Perspective
“Adj.” means segment adjacent from Angle of Perspective

7. If the Angle of Perspective is
then
then
Hyp
Adj
Hyp
Opp
Opp
Adj

8. The 3 Trigonometric Ratios
The 3 ratios are Sine, Cosine and Tangent

9. The Amazing Legend of…
Chief SohCahToa

10. Chief SohCahToa
Once upon a time there was a wise old Native American Chief named Chief SohCahToa.
He was named that due to a 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.

11. Trigonometric Ratios
To 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
Sin A = Opposite side        SOH             Hypotenuse
Cos A = Adjacent side         CAH            Hypotenuse
Tan A = Opposite side    TOA                   Adjacent side c b a C B

12. SohCahToa

13.  the side opposite the acute angle , opp
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: θ
hyp
 the side opposite the acute angle ,
opp
 the side adjacent to the acute angle ,
 and the hypotenuse of the right triangle.
adj
The trigonometric functions are
sine, cosine, tangent, cotangent, secant, and cosecant.
sin 𝜃= 𝑜𝑝𝑝 ℎ𝑦𝑝
cos 𝜃= 𝑎𝑑𝑗 ℎ𝑦𝑝
tan 𝜃= 𝑜𝑝𝑝 𝑎𝑑𝑗
csc 𝜃= ℎ𝑦𝑝 𝑜𝑝𝑝
sec 𝜃= ℎ𝑦𝑝 𝑎𝑑𝑗
cot 𝜃= 𝑎𝑑𝑗 𝑜𝑝𝑝

14. 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 𝑥 2 + 𝑦 2 =1

15. 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.

16. 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.

17. The Unit Circle
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
 is the angle of rotation
(𝒄𝒐𝒔, 𝐬𝐢𝐧 )
(𝒙, 𝒚) 1 y
cos 𝜃= 𝑎𝑑𝑗 ℎ𝑦𝑝 = 𝑥 1 =𝑥 x
sin 𝜃= 𝑜𝑝𝑝 ℎ𝑦𝑝 = 𝑦 1 =𝑦

18. The Unit Circle
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.
 is the angle of rotation
cos 𝜃=𝑥
(𝒄𝒐𝒔, 𝐬𝐢𝐧 )
𝑠𝑖𝑛 𝜃=𝑦
(𝒙, 𝒚) 1 y x

19. Trigonometric Functions
 is the angle of rotation 1 y x
These functions are reciprocals of each other.

20. 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. II I
III IV

21. The Unit Circle Completion
sin
One 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. 1
The x-axis corresponds to the cosine function, and the y-axis corresponds to the sine function.
cos
In order to complete our unit circle with the missing coordinates, we must use the special right triangles below:
45º 1
30º
60º 1
30º -60º -90º
45º -45º -90º
The hypotenuse for each triangle is 1 unit.

22. Use the Pythagorean Theorem to help find the sides.
You first need to find the lengths of the other sides of each right triangle...
Use the Pythagorean Theorem to help find the sides.
In a triangle, the shortest side is of the hypotenuse.
30º
60º 1
Find the remaining side.
45º 1
The two legs of a 45 – 45 – 90 triangle have the same length. Use the Pythagorean Theorem to find their sides.

23. Usefulness of Knowing Trigonometric Functions of Special Angles: 30o, 45o, 60o
The trigonometric function values derived from knowing the side ratios of the and 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

24. Now, use the corresponding triangle to find the coordinates on the unit circle...
(0, 1)
sin
What are the
coordinates of
this point?
(Use your
triangle)
(–1, 0)
30º
cos
(1, 0)
(0, –1)

25. Now, use the corresponding triangle to find the coordinates on the unit circle...
(0, 1)
sin
What are the
coordinates of
this point?
(Use your
triangle)
(–1, 0)
45º
cos
(1, 0)
(0, –1)

26. You can use your special right triangles to find any of the points on the unit circle...
(0, 1)
sin
( 1 2 , ) 
(Use your
triangle)
What are the
coordinates of
this point?
(–1, 0)
cos
(1, 0)
(0, –1)

27. You can use your special right triangles to find any of the points on the unit circle...
(0, 1)
sin
( 1 2 , )
Notice the coordinates in quadrants I and III.
(–1, 0)
cos
(1, 0)
(Use your
triangle)
What are the
coordinates of
this point?
(0, –1)

28. Now we can complete the unit circle with the coordinates.
(0, 1)
sin
( 1 2 , )
(–1, 0)
cos
(1, 0)
(0, –1)

29. Let’s look at a quick way to get the coordinates (in-class only)
(0, 1)
(–1, 0)
(1, 0)
(0, –1)

30. You should memorize this
You should memorize this. This is a great reference because you can figure out the trig functions of all these angles quickly. 2𝜋
360 °

31. Here is the unit circle divided into 8 pieces
Here is the unit circle divided into 8 pieces. Can you figure out how many degrees are in each division?
These are easy to memorize since they all have the same value with different signs depending on the quadrant.
90°
135°
45°
180°
45° 0°
225°
315°
270°
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.

32. Can you figure out what these angles would be in radians?
90°
135°
45°
180° 0°
225°
315°
270°
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.

33. You'll need to memorize these too but you can see the pattern.
Here is the unit circle divided into 12 pieces. Can you figure out how many degrees are in each division?
You'll need to memorize these too but you can see the pattern.
90°
120°
60°
150°
30°
180°
30° 0°
210°
330°
240°
300°
270°
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.

34. We'll see them all put together on the unit circle on the next screen.
Can you figure out what the angles would be in radians?
We'll see them all put together on the unit circle on the next screen.
90°
120°
60°
150°
30°
180°
30° 0°
210°
330°
240°
300°
270°
It is still  halfway around the circle and the upper half is divided into 6 pieces so each piece is /6.

35. You should memorize this
You should memorize this. This is a great reference because you can figure out the trig functions of all these angles quickly. 2𝜋
360 °

36. 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 𝑦 𝑥 .
(0,1)
(-1,0)
(1,0) 
(0,-1)
We divide the unit circle into various pieces and learn the point values so we can then from memory find trig functions.

37. We know all of the sides of this triangle
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.
3 2
(0,1)
1 2
(-1,0)
(1,0) 
(0,-1)
Notice the sine is just the y value of the unit circle point and the cosine is just the x value.

38. Finding Values of the Trigonometric Functions
Find the values of the six trig functions at 𝜃= 𝜋 2
What are the coordinates?
(0,1)
1 𝑦
csc 𝜋 2 =
sin 𝜋 2 = =1 𝑦 =1
sec 𝜋 2 =
1 𝑥
cos 𝜋 2 =
=𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑥 =0
tan 𝜋 2 =
𝑦 𝑥
=𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
cot 𝜋 2 =
𝑥 𝑦 =0

39. Finding Values of the Trigonometric Functions
Find the values of the six trig functions at 𝜃= 𝜋 4
( , )
What are the coordinates? =
1 𝑦 = = 2
csc 𝜋 4 =
sin 𝜋 4 = 𝑦
2 2
cos 𝜋 4 =
sec 𝜋 4 =
1 𝑥 = = 2
𝑥 =
tan 𝜋 4 =
𝑦 𝑥
cot 𝜋 4 =
𝑥 𝑦 =1 =1

40. Let’s look at an applet.
What is the domain of sine and cosine?
All real numbers
What is the range of sine and cosine?
[−1, 1]

41. Even and Odd Trig Functions
The cosine and secant functions are even functions.
cos −𝜃 = cos 𝜃 sec −𝜃 = sec 𝜃
The sine, cosecant, tangent, and cotangent functions are odd.
sin −𝜃 =− sin 𝜃
csc −𝜃 =− csc 𝜃
tan −𝜃 =− tan 𝜃
cot −𝜃 =− cot 𝜃

42. This is an even function. Think “same as”.
Remember negative angle means to go clockwise

43. This is an odd function. Think “opposite”.

44. 𝐭𝐚𝐧 −𝜽=− 𝐭𝐚𝐧 𝜽
This is an odd function.

45. Using Even and Odd Functions to Find Values of Trig Functions
Find the value of:
cos (− 𝜋 4 )
Is cosine an even or odd function? It is even. It has the same value as cos⁡( 𝜋 4 ), which is the x-coordinate for 𝜋 4 , and that is
Answer: cos − 𝜋 4 = cos 𝜋 4 =

46. Using Even and Odd Functions to Find Values of Trig Functions
Find the value of:
tan (− 𝜋 4 )
Is tangent an even or odd function? It is odd. It has the opposite sign value as tan⁡( 𝜋 4 ), which is −1.
Answer: tan − 𝜋 4 = −tan 𝜋 4 = −1

47. Reciprocal Identities

48. Quotient Identities
tan 𝜃 = sin θ cos θ
cot 𝜃 = cos θ sin θ

49. Using Quotient and Reciprocal Identities
Given sin 𝜃= and cos 𝜃 = , find the value of each of the four remaining trig functions.
We need to find tangent, cotangent, secant, and cosecant.
tan 𝜃 = sin θ cos θ
Find the remaining three. Express the answer in exact form.
tan 𝜃= =

50. Using Quotient and Reciprocal Identities
Given sin 𝜃= and cos 𝜃= , find the value of each of the four remaining trig functions.
Now we need to find cotangent.
cot 𝜃 = 𝑐𝑜𝑠 θ sin θ
Finish out the problem. =
21 2

51. Using Quotient and Reciprocal Identities
Given sin 𝜃= and cos 𝜃= , find the value of each of the four remaining trig functions.
Now we need to find secant.
sec θ= 1 cos θ
Finish out the problem. =

52. Using Quotient and Reciprocal Identities
Given sin 𝜃= and cos 𝜃= , find the value of each of the four remaining trig functions.
Now we need to find cosecant.
csc θ= 1 𝑠𝑖𝑛 θ
1 2 5
Finish out the problem.
= 5 2

53. Pythagorean Identities
The equation of a unit circle is 𝑥 2 + 𝑦 2 =1
Since cos 𝜃=𝑥 and sin θ=y, then
𝒔𝒊𝒏 𝟐 𝜽+ 𝒄𝒐𝒔 𝟐 𝜽=𝟏

54. Pythagorean Identities
𝐬𝐢𝐧 𝟐 𝜽+ 𝐜𝐨𝐬 𝟐 𝜽=𝟏
𝒔𝒊 𝒏 𝟐 𝜽 𝒔𝒊 𝒏 𝟐 𝜽 + 𝒄𝒐 𝒔 𝟐 𝜽 𝒔𝒊 𝒏 𝟐 𝜽
= 𝟏 𝒔𝒊 𝒏 𝟐 𝜽
𝟏+𝒄𝒐 𝒕 𝟐 𝜽=𝒄𝒔 𝒄 𝟐 𝜽

55. Pythagorean Identities
𝐬𝐢𝐧 𝟐 𝛉+ 𝐜𝐨𝐬 𝟐 𝛉=𝟏
𝒔𝒊 𝒏 𝟐 𝜽 𝒄𝒐 𝒔 𝟐 𝜽 + 𝒄𝒐 𝒔 𝟐 𝜽 𝒄𝒐 𝒔 𝟐 𝜽 =
𝟏 𝒄𝒐 𝒔 𝟐 𝜽
𝒕𝒂 𝒏 𝟐 𝜽+𝟏=𝒔𝒆 𝒄 𝟐 𝜽

56. Using a Pythagorean Identity
Given that sin 𝜃= and 0≤𝜃< 𝜋 2 , find the value of cos 𝜃 using a trig identity.
We can find the value of cos 𝜃 using the Pythagorean Identity.
sin 2 𝜃+ cos 2 𝜃= cos 2 𝜃=1
cos 2 𝜃= cos 2 𝜃=1− 9 25
cos 2 𝜃= 16 25
cos 𝜃 = = 4 5

57. Periodic Functions
A 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 𝜋 radians. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena.
A period of 2𝜋 is one revolution around the unit circle. A period of 𝜋 is one-half revolution.

58. Periodic Properties of the Sine and Cosine Functions
sin 𝜃+2𝜋 = sin 𝜃
and
cos 𝜃+2𝜋 = cos 𝜃
The sine and cosine functions are periodic functions and have a period of 2𝜋.
The secant and cosecant functions are also periodic functions and have a period of 2𝜋.

59. Periodic Properties of the Tangent and Cotangent Functions
cot 𝜃+𝜋 = cot 𝜃
The tangent and cotangent functions are periodic functions and have a period of 𝜋.

60. Sine and cosine are periodic with a period of 360 or 2.
Let's label the unit circle with values of the tangent. (Remember this is just y/x)
We see that they repeat every  so the tangent’s period is .

61. 1 PERIODIC PROPERTIES sin( + 2) = sin  csc( + 2) = csc 
Reciprocal functions have the same period.
PERIODIC PROPERTIES
sin( + 2) = sin  csc( + 2) = csc 
cos( + 2) = cos  sec( + 2) = sec 
tan( + ) = tan  cot( + ) = cot 
This would have the same value as 1
(you can count around on unit circle or subtract the period twice.)

62. Examples:
Evaluate the trigonometric function using its period as an aid
= cos 5𝜋− cos 2𝜋=
cos 3𝜋 − cos 2𝜋=
cos 𝜋= −1
= sin 9𝜋 4 − sin 8𝜋 4 =
2 2
sin 𝜋 4 =

63. Examples:
Evaluate the trigonometric function using its period as an aid
= cos − 8𝜋 3 + cos 6𝜋 3
= cos − 2𝜋 3 + cos 6𝜋 3
= cos 4𝜋 3
− 1 2
= sin 19𝜋 6 − sin 12𝜋 6
= sin 7𝜋 6
=− 1 2