This document provides an introduction to trigonometry and right triangle trigonometry. It defines the six trigonometric ratios (sine, cosine, tangent, cosecant, secant, and cotangent) and relates them to the sides of a right triangle. It discusses measuring angles in degrees and converting between degree-minute-second and decimal degree formats. It introduces the trigonometric functions of 30°, 45°, and 60° degrees and explains trigonometric identities and properties for 30-60-90 and 45-45-90 right triangles. The document provides examples of evaluating trig functions using right triangles and calculators.

1. 6-1: Right-Triangle
Trigonometry
© 2007 Roy L. Gover (www.mrgover.com)
Learning Goals:
•Define the six trig ratios of
an acute angle
•Evaluate trig ratios using
triangles, a calculator and
special angles
•Introduce trigonometry and
angle measurement

2. Important Idea
Trigonometry, which means
triangle measurement was
developed by the Greeks in
the 2nd
century BC. It was
originally used only in
astronomy, navigation and
surveying. It is now used to
model periodic behavior such
as sound waves and
planetary orbits.

3. Definition
Angle: the figure formed by
two rays with a common
endpoint

4. Definition
Initial Side
Terminal Side
Verte
x
Angle A
is in
standard
position
A
x
y

5. 43
1
Definition
x
yThe
Cartesian
Plane is
divided into
quadrants
as follows:
2

6. Important Idea
Angles may be measured in
degrees where 1 degree (°)
is 1/360 of a circle, 90° is ¼
of a circle, 180° is ½ of a
circle, 270° is ¾ of a circle
and 360° is a full circle. A 90°
angle is also called a right
angle.

7. Example
90°
180°
270°
0° or 360°

8. Example
90°
180°
270°
0° or 360°

9. Example
90°
180°
270°
0° or 360°

10. Example
90°
180°
270°
0° or 360°

11. Example
90°
180°
270°
0° or 360°

12. Try This What is the
measure of this
angle?
a. 0°
b. 45°
c. 90°
d. 120°
e. 180°

13. Try This What is the
measure of this
angle?
a. 0°
b. 45°
c. 90°
d. 120°
e. 180°

14. Try This What is the
measure of this
angle?
a. 0°
b. 45°
c. 90°
d. 120°
e. 180°

15. Try This What is the
measure of this
angle?
a. 0°
b. 45°
c. 90°
d. 120°
e. 180°

16. Try This What is the
measure of this
angle?
a. 0°
b. 45°
c. 90°
d. 120°
e. 180°

17. Definition
Fractional parts of a degree
can be written in decimal
form or Degree-Minute –
Second (DMS) form. A
minute is 1/60 of a degree. A
second is 1/60 of a minute.
How many seconds are in
each degree?

18. Example
Write 29°40’20” in decimal
degrees accurate to 3
decimal places.
Symbol for
minute
Symbol
for second

19. Important Idea
To convert from DMS to
decimal, write the decimal
expression 29°40’20” as:
40 20
29
60 3600
+ +
in your calculator.

20. Try This
77.399°
Write
77°23’56’’
in decimal
form.
0°
90°
180°
270°

21. Try This
185.751°
Write
185°45’3’’
in decimal
form.
0°
90°
180°
270°

22. Try This
319.541°
Write
319°32’28’’
in decimal
degrees
accurate to
3 decimal
places.
0°
90°
180°
270°

23. Example
Write
37.576°
in DMS
form.
Procedure:
1. Write the decimal
part .576° as .576 x
60=34.56’
2. Write the decimal
part of minutes as .
56 x 60=33.6’’
3. Round 33.6’’ to
34’’ for total
37°34’34’’

24. Try This
Write
185.651°
in DMS
form.
0°
90°
180°
270°185°39’4’’

25. Try This
Write
85.259°
in DMS
form.
0°
90°
180°
270°85°15’32’’

26. Definition
A
B
C
a
b
c
D
E
F
d
e
f
m A m D∠ = ∠If then
a d
c f
=
b e
c f
=
a d
b e
=& &
Similar
Triangles

27. Important Idea
If 2 right triangles have
equal angles, the
corresponding ratios of
their sides must be the
same no matter the size of
the triangles. This fact is
the basis for trigonometry.

28. Example
A
B
C
a
b
c
D
E
F
d
e
f
30m A m D∠ = ∠ = °If and
2, 4a c= = and 3d = then ?f =

29. Try This
A B
C
ab
c D
E
F
d
e
f
60m A m D∠ = ∠ = °If and
2, 4c b= = and 6e = then ?f =
12

30. Definition
The hypotenuse
is the side
opposite the 90° angle and
is the longest side. The
other 2 sides are legs.

31. Definition
The
opposite
side
is the leg
opposite
the given
angle
A
C

32. Definition
The
adjacent
side
is the leg
next to the
given angle
(not the
hypotenuse).
A
C

33. Important Idea
Right
triangles
come in all
sizes,
shapes and
orientations.

34. Definition
For a given acute angle in
a right triangle:
θ
The sine of written as
is the ratio
θ sinθ
sinθ =
opposite
hypotenuse
(see p.416 of your text):
M
em
orize

35. Definition
For a given acute angle in
a right triangle:
θ
The cosine of written asθ
cosθ
cosθ =
adjacent
hypotenuse
is the ratio:
(see p.416 of your text):
M
em
orize

36. Definition
For a given acute angle in
a right triangle:
θ
The tangent of written asθ
tanθ
tanθ =
opposite
adjacent
is the ratio:
(see p.416 of your text):
M
em
orize

37. opposite
Definition
For a given acute angle in
a right triangle:
θ
The cosecant of written asθ
cscθ
1
csc
sin
θ
θ
= = hypotenuse
is the ratio:
(see p.416 of your text):
M
em
orize

38. adjacent
Definition
For a given acute angle in
a right triangle:
θ
The secant of written asθ
secθ
1
sec
cos
θ
θ
= = hypotenuse
is the ratio:
(see p.416 of your text):
M
em
orize

39. opposit
e
Definition
For a given acute angle in
a right triangle:
θ
The cotangent of written
as
θ
cotθ
1
cot
tan
θ
θ
= = adjacent
is the ratio:
(see p.416 of your text):
M
em
orize

40. Example
θ
13
5
12
Evaluate
the 6 trig
ratios of
the angleθ

41. Try This
θ
35
4
Evaluate
the 6 trig
ratios of
the angle θ
4
sin
5
θ =
3
cos
5
θ =
4
tan
3
θ =
5
csc
4
θ =
5
sec
3
θ =
3
cot
4
θ =

42. Try This
θ13
5
12
Evaluate
the 6 trig
ratios of
the angle θ
5
sin
13
θ =
12
cos
13
θ =
5
tan
12
θ =
13
csc
5
θ =
13
sec
12
θ =
12
cot
5
θ =

43. Example
Using your
calculator,
evaluate the 6
trig ratios of
33°
Be sure that
mode is set to
degrees

44. Try This
Using your calculator,
evaluate the 6 trig ratios of
117.25°
sin117.25 .889° =
cos117.25 .458° = −
tan117.25 1.942° = −
csc117.25 1.125=
sec117.25 2.184= −
cot117.25 .515= −

45. Try This
Using your calculator,
evaluate cos12 15'30''°
cos12 15'30'' cos12.258 .977° = =

46. Important Idea
1
csc
sin
θ
θ
=
1
sec
cos
θ
θ
=
Since your
calculator
does not have
a sec, csc or
cot key, you
must find the
reciprocal of
cos, sin or tan.
1
cot
tan
θ
θ
=

47. Definition
The special angles are:
•30°
•60°
•45°

48. Important Idea
These angles are special
because they have exact
value trig functions.

49. Consider the first two
special angles in degrees...
30°
60°
Long Side ShortSide
Hypotenuse
Analysis

50. Important Idea
In a 30°-60°-90° right
triangle, the short side is
opposite the 30° angle, the
long side is opposite the 60°
angle, and the hypotenuse
is opposite the 90° angle.

51. LongSide
Hypotenuse
Short Side…orientation
does not
change the
relationships
between
sides and
angles
60°
30°
Important Idea

52. LongSide
Hypotenuse
Short Side
…orientation
does not
change the
relationships
between
sides and
angles
30°
60°
Important Idea

53. Important Idea
In a 30°,60°,90°
triangle:
•the short side is one-half
the hypotenuse.
•the long side is times
the short side.
3Memoriz
e

54. Try This
Find the
length of
the
missing
sides: 30°
60°
4
8
4 3

55. Try This
Find the
length of
the
missing
sides:
10
5
5 3
30°

56. Try This
Find the
length of
the
missing
sides:
5
5 3
3
10 3
3
60°

57. Try This
Find the
length of
the
missing
sides:
60°
4
8
4 3

58. 30°
60°
45°
45°
45°
Analysis
Consider the
last special
angle:

59. Hypotenuse…orientation
does not
change the
relationships
between
sides and
angles
45°
45°
Important Idea

60. H
ypotenuse
…orientation
does not
change the
relationships
between
sides and
angles
45°
45°
Important Idea

61. …and sides
opposite
equal
angles are
equal...
x
x
and by the
2x
pythagorean theorem, the
hypotenuse is...
45°
45°
Important Idea

62. Important Idea
In a 45°,45°,90°
triangle:
•The legs of the triangle
are equal.
•the hypotenuse is
times the length of the leg.
Memoriz
e
2

63. Try This
Find
the
length
of the
missing
sides 2
2
2 2
45°

64. Try This
Find
the
length
of the
missing
sides
2
2
2
45°
45°

65. Example
Find the
exact value
of the 6 trig
functions of
30°. This is
not a
calculator
problem.
30°

66. Example
Find the
exact value
of the 6 trig
functions of
30°. This is
not a
calculator
problem.
30°

67. Important Idea
When you know the
lengths of the 3 sides of
a right triangle, you can
evaluate any of the 6 trig
functions.

68. Example
Find the
exact value
of the 6 trig
functions of
45°. This is
not a
calculator
problem.
45°

69. Try This
Find the exact
value of the 6 trig
functions of 60°.
Do not use a
calculator.
60°

70. Solution
60°
1
2
3
3
sin60
2
° =
1
cos60
2
° =
tan60 3° =

71. Solution
60°
1
2
3
2 3
csc60
3
° =
sec60 2° =
3
cot60
3
° =

72. Lesson Close
We will use the information in
this lesson to solve right
triangle problems in the next
lesson. Right triangle
problems are used in real-
world applications such as
indirect measurement,
surveying and navigation.