6-1: Right-Triangle
Trigonometry
© 2007 Roy L. Gover (www.mrgover.com)
Learning Goals:
•Define the six trig ratios of
an a...
Important Idea
Trigonometry, which means
triangle measurement was
developed by the Greeks in
the 2nd
century BC. It was
or...
Definition
Angle: the figure formed by
two rays with a common
endpoint
Definition
Initial Side
Terminal Side
Verte
x
Angle A
is in
standard
position
A
x
y
43
1
Definition
x
yThe
Cartesian
Plane is
divided into
quadrants
as follows:
2
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 ...
Example
90°
180°
270°
0° or 360°
Example
90°
180°
270°
0° or 360°
Example
90°
180°
270°
0° or 360°
Example
90°
180°
270°
0° or 360°
Example
90°
180°
270°
0° or 360°
Try This What is the
measure of this
angle?
a. 0°
b. 45°
c. 90°
d. 120°
e. 180°
Try This What is the
measure of this
angle?
a. 0°
b. 45°
c. 90°
d. 120°
e. 180°
Try This What is the
measure of this
angle?
a. 0°
b. 45°
c. 90°
d. 120°
e. 180°
Try This What is the
measure of this
angle?
a. 0°
b. 45°
c. 90°
d. 120°
e. 180°
Try This What is the
measure of this
angle?
a. 0°
b. 45°
c. 90°
d. 120°
e. 180°
Definition
Fractional parts of a degree
can be written in decimal
form or Degree-Minute –
Second (DMS) form. A
minute is 1...
Example
Write 29°40’20” in decimal
degrees accurate to 3
decimal places.
Symbol for
minute
Symbol
for second
Important Idea
To convert from DMS to
decimal, write the decimal
expression 29°40’20” as:
40 20
29
60 3600
+ +
in your cal...
Try This
77.399°
Write
77°23’56’’
in decimal
form.
0°
90°
180°
270°
Try This
185.751°
Write
185°45’3’’
in decimal
form.
0°
90°
180°
270°
Try This
319.541°
Write
319°32’28’’
in decimal
degrees
accurate to
3 decimal
places.
0°
90°
180°
270°
Example
Write
37.576°
in DMS
form.
Procedure:
1. Write the decimal
part .576° as .576 x
60=34.56’
2. Write the decimal
par...
Try This
Write
185.651°
in DMS
form.
0°
90°
180°
270°185°39’4’’
Try This
Write
85.259°
in DMS
form.
0°
90°
180°
270°85°15’32’’
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
Important Idea
If 2 right triangles have
equal angles, the
corresponding ratios of
their sides must be the
same no matter ...
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 =
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
Definition
The hypotenuse
is the side
opposite the 90° angle and
is the longest side. The
other 2 sides are legs.
Definition
The
opposite
side
is the leg
opposite
the given
angle
A
C
Definition
The
adjacent
side
is the leg
next to the
given angle
(not the
hypotenuse).
A
C
Important Idea
Right
triangles
come in all
sizes,
shapes and
orientations.
Definition
For a given acute angle in
a right triangle:
θ
The sine of written as
is the ratio
θ sinθ
sinθ =
opposite
hypot...
Definition
For a given acute angle in
a right triangle:
θ
The cosine of written asθ
cosθ
cosθ =
adjacent
hypotenuse
is the...
Definition
For a given acute angle in
a right triangle:
θ
The tangent of written asθ
tanθ
tanθ =
opposite
adjacent
is the ...
opposite
Definition
For a given acute angle in
a right triangle:
θ
The cosecant of written asθ
cscθ
1
csc
sin
θ
θ
= = hypo...
adjacent
Definition
For a given acute angle in
a right triangle:
θ
The secant of written asθ
secθ
1
sec
cos
θ
θ
= = hypote...
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
= = a...
Example
θ
13
5
12
Evaluate
the 6 trig
ratios of
the angleθ
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
co...
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...
Example
Using your
calculator,
evaluate the 6
trig ratios of
33°
Be sure that
mode is set to
degrees
Try This
Using your calculator,
evaluate the 6 trig ratios of
117.25°
sin117.25 .889° =
cos117.25 .458° = −
tan117.25 1.94...
Try This
Using your calculator,
evaluate cos12 15'30''°
cos12 15'30'' cos12.258 .977° = =
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 th...
Definition
The special angles are:
•30°
•60°
•45°
Important Idea
These angles are special
because they have exact
value trig functions.
Consider the first two
special angles in degrees...
30°
60°
Long Side ShortSide
Hypotenuse
Analysis
Important Idea
In a 30°-60°-90° right
triangle, the short side is
opposite the 30° angle, the
long side is opposite the 60...
LongSide
Hypotenuse
Short Side…orientation
does not
change the
relationships
between
sides and
angles
60°
30°
Important Id...
LongSide
Hypotenuse
Short Side
…orientation
does not
change the
relationships
between
sides and
angles
30°
60°
Important I...
Important Idea
In a 30°,60°,90°
triangle:
•the short side is one-half
the hypotenuse.
•the long side is times
the short si...
Try This
Find the
length of
the
missing
sides: 30°
60°
4
8
4 3
Try This
Find the
length of
the
missing
sides:
10
5
5 3
30°
Try This
Find the
length of
the
missing
sides:
5
5 3
3
10 3
3
60°
Try This
Find the
length of
the
missing
sides:
60°
4
8
4 3
30°
60°
45°
45°
45°
Analysis
Consider the
last special
angle:
Hypotenuse…orientation
does not
change the
relationships
between
sides and
angles
45°
45°
Important Idea
H
ypotenuse
…orientation
does not
change the
relationships
between
sides and
angles
45°
45°
Important Idea
…and sides
opposite
equal
angles are
equal...
x
x
and by the
2x
pythagorean theorem, the
hypotenuse is...
45°
45°
Importan...
Important Idea
In a 45°,45°,90°
triangle:
•The legs of the triangle
are equal.
•the hypotenuse is
times the length of the ...
Try This
Find
the
length
of the
missing
sides 2
2
2 2
45°
Try This
Find
the
length
of the
missing
sides
2
2
2
45°
45°
Example
Find the
exact value
of the 6 trig
functions of
30°. This is
not a
calculator
problem.
30°
Example
Find the
exact value
of the 6 trig
functions of
30°. This is
not a
calculator
problem.
30°
Important Idea
When you know the
lengths of the 3 sides of
a right triangle, you can
evaluate any of the 6 trig
functions.
Example
Find the
exact value
of the 6 trig
functions of
45°. This is
not a
calculator
problem.
45°
Try This
Find the exact
value of the 6 trig
functions of 60°.
Do not use a
calculator.
60°
Solution
60°
1
2
3
3
sin60
2
° =
1
cos60
2
° =
tan60 3° =
Solution
60°
1
2
3
2 3
csc60
3
° =
sec60 2° =
3
cot60
3
° =
Lesson Close
We will use the information in
this lesson to solve right
triangle problems in the next
lesson. Right triangl...
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  • .576 x 60=34.56’; .56 x 60=33.6’’ round to 34 ‘’: 37deg34min34sec
  • Hprec6 1

    1. 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. 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. 3. Definition Angle: the figure formed by two rays with a common endpoint
    4. 4. Definition Initial Side Terminal Side Verte x Angle A is in standard position A x y
    5. 5. 43 1 Definition x yThe Cartesian Plane is divided into quadrants as follows: 2
    6. 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. 7. Example 90° 180° 270° 0° or 360°
    8. 8. Example 90° 180° 270° 0° or 360°
    9. 9. Example 90° 180° 270° 0° or 360°
    10. 10. Example 90° 180° 270° 0° or 360°
    11. 11. Example 90° 180° 270° 0° or 360°
    12. 12. Try This What is the measure of this angle? a. 0° b. 45° c. 90° d. 120° e. 180°
    13. 13. Try This What is the measure of this angle? a. 0° b. 45° c. 90° d. 120° e. 180°
    14. 14. Try This What is the measure of this angle? a. 0° b. 45° c. 90° d. 120° e. 180°
    15. 15. Try This What is the measure of this angle? a. 0° b. 45° c. 90° d. 120° e. 180°
    16. 16. Try This What is the measure of this angle? a. 0° b. 45° c. 90° d. 120° e. 180°
    17. 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. 18. Example Write 29°40’20” in decimal degrees accurate to 3 decimal places. Symbol for minute Symbol for second
    19. 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. 20. Try This 77.399° Write 77°23’56’’ in decimal form. 0° 90° 180° 270°
    21. 21. Try This 185.751° Write 185°45’3’’ in decimal form. 0° 90° 180° 270°
    22. 22. Try This 319.541° Write 319°32’28’’ in decimal degrees accurate to 3 decimal places. 0° 90° 180° 270°
    23. 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. 24. Try This Write 185.651° in DMS form. 0° 90° 180° 270°185°39’4’’
    25. 25. Try This Write 85.259° in DMS form. 0° 90° 180° 270°85°15’32’’
    26. 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. 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. 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. 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. 30. Definition The hypotenuse is the side opposite the 90° angle and is the longest side. The other 2 sides are legs.
    31. 31. Definition The opposite side is the leg opposite the given angle A C
    32. 32. Definition The adjacent side is the leg next to the given angle (not the hypotenuse). A C
    33. 33. Important Idea Right triangles come in all sizes, shapes and orientations.
    34. 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. 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. 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. 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. 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. 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. 40. Example θ 13 5 12 Evaluate the 6 trig ratios of the angleθ
    41. 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. 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. 43. Example Using your calculator, evaluate the 6 trig ratios of 33° Be sure that mode is set to degrees
    44. 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. 45. Try This Using your calculator, evaluate cos12 15'30''° cos12 15'30'' cos12.258 .977° = =
    46. 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. 47. Definition The special angles are: •30° •60° •45°
    48. 48. Important Idea These angles are special because they have exact value trig functions.
    49. 49. Consider the first two special angles in degrees... 30° 60° Long Side ShortSide Hypotenuse Analysis
    50. 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. 51. LongSide Hypotenuse Short Side…orientation does not change the relationships between sides and angles 60° 30° Important Idea
    52. 52. LongSide Hypotenuse Short Side …orientation does not change the relationships between sides and angles 30° 60° Important Idea
    53. 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. 54. Try This Find the length of the missing sides: 30° 60° 4 8 4 3
    55. 55. Try This Find the length of the missing sides: 10 5 5 3 30°
    56. 56. Try This Find the length of the missing sides: 5 5 3 3 10 3 3 60°
    57. 57. Try This Find the length of the missing sides: 60° 4 8 4 3
    58. 58. 30° 60° 45° 45° 45° Analysis Consider the last special angle:
    59. 59. Hypotenuse…orientation does not change the relationships between sides and angles 45° 45° Important Idea
    60. 60. H ypotenuse …orientation does not change the relationships between sides and angles 45° 45° Important Idea
    61. 61. …and sides opposite equal angles are equal... x x and by the 2x pythagorean theorem, the hypotenuse is... 45° 45° Important Idea
    62. 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. 63. Try This Find the length of the missing sides 2 2 2 2 45°
    64. 64. Try This Find the length of the missing sides 2 2 2 45° 45°
    65. 65. Example Find the exact value of the 6 trig functions of 30°. This is not a calculator problem. 30°
    66. 66. Example Find the exact value of the 6 trig functions of 30°. This is not a calculator problem. 30°
    67. 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. 68. Example Find the exact value of the 6 trig functions of 45°. This is not a calculator problem. 45°
    69. 69. Try This Find the exact value of the 6 trig functions of 60°. Do not use a calculator. 60°
    70. 70. Solution 60° 1 2 3 3 sin60 2 ° = 1 cos60 2 ° = tan60 3° =
    71. 71. Solution 60° 1 2 3 2 3 csc60 3 ° = sec60 2° = 3 cot60 3 ° =
    72. 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.

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