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Name ________________________________________ Date __________________ Class__________________
Reteach
LESSON
10-1
Right-Angle Trigonometry
A trigonometric ratio compares the lengths of two sides of a right triangle.
The values of the ratios depend upon one of the acute angles of the
triangle, denoted by θ.
Use SOHCAHTOA to remember the
relationships between the sides of a right
Sine is Opposite over Hypotenuse,
Cosine is Adjacent over Hypotenuse,
triangle that correspond to the trigonometric
Tangent is Opposite over Adjacent.
ratios sine, cosine, and tangent.
a
Opposite
=
Hypotenuse c
sin θ =
cos θ =
b
Adjacent
=
Hypotenuse c
tan θ =
Opposite a
=
Adjacent b
Use the definitions of each ratio and the corresponding values from a given right triangle to
find the values of the trigonometric functions for θ.
Opposite
14
7
=
=
Hypotenuse 50 25
sin θ =
cos θ =
Adjacent
48 24
=
=
Hypotenuse 50 25
tan θ =
Opposite 14
7
=
=
Adjacent 48 24
Find the value of the sine, cosine, and tangent functions for θ.
1.
2.
3.
Opposite
= ____________
Hypotenuse
sinθ = ____________
sinθ = ____________
cos θ =
Adjacent
= ____________
Hypotenuse
cosθ = ____________
cosθ = ____________
tan θ =
Opposite
= ____________
Adjacent
tanθ = ____________
tanθ = ____________
sin θ =
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
10-6
Holt McDougal Algebra 2
Name ________________________________________ Date __________________ Class__________________
LESSON
10-1
Reteach
Right-Angle Trigonometry (continued)
The reciprocals of the sine, cosine, and tangent ratios are
also trigonometric ratios.
The cosecant function (csc θ ) is the
reciprocal of the sine function.
csc θ =
1
Hypotenuse c
=
=
sin θ
Opposite
a
The secant function (sec θ ) is the
reciprocal of the cosine function.
sec θ =
1
Hypotenuse c
=
=
cos θ
Adjacent
b
The cotangent function (cot θ ) is the
reciprocal of the tangent function.
cot θ =
1
Adjacent b
=
=
tan θ Opposite a
Use the reciprocal relationship of the ratios to find the values of the
reciprocal trigonometric functions.
sin θ =
3
5
csc θ =
1
5
=
sin θ
3
cos θ =
4
5
sec θ =
1
5
=
cos θ 4
tan θ =
3
4
cot θ =
1
4
=
tan θ
3
Find the values of the six trigonometric functions for θ.
4.
5.
sin θ = ___________ csc θ =
1
= ________________
sin θ
cos θ = ___________ sec θ =
1
= ________________
cos θ
tan θ = ___________ cot θ =
1
= _________________
tan θ
______________
__________________
______________
__________________
______________
__________________
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
10-7
Holt McDougal Algebra 2
10-1 RIGHT-ANGLE TRIGONOMETRY
10. 47 ft
11. 162 ft
12. 4817 m
Practice A
1. a. sinθ =
48 24
=
50 25
b. cos θ =
14
7
=
50 25
Practice C
48 24
=
c. tanθ =
14
7
2.
3 4 3
, ,
5 5 4
4.
12 5 12
, ,
13 13 5
1.
7 24 7
;
;
25 25 24
2.
3.
60 11 60
; ;
61 61 11
4. 20 3
5. 3.4 3
3.
9 40 9
,
,
41 41 40
b. cos 45° =
7
24
; cos θ =
;
25
25
tanθ =
7
25
; csc θ =
;
24
7
sec θ =
25
24
; cot θ =
24
7
c. cos 45° =
x
12 2
2
2
8. sinθ =
12
5
; cos θ =
;
13
13
tan θ =
12
13
; csc θ =
;
5
12
sec θ =
13
5
; cot θ =
5
12
d. x = 12
6. 10
7. 8
8. 9
9. 45 ft
6. 16
7. sinθ =
5.a. Cosine
9. sinθ =
Practice B
1.
4 3 4
; ;
5 5 3
2.
9 40 9
;
;
41 41 40
3.
12 5 12
; ;
13 13 5
4. 6 3
5.
44 3
3
6. 7
12
5
12
; cos θ =
; tan θ =
7. sin θ =
13
13
5
4 3 4
; ;
5 5 3
10
15
; cos θ =
;
5
5
tan θ =
6
10
; csc θ =
;
3
2
sec θ =
15
6
; cot θ =
3
2
10. 83.5 ft
11. 287 ft
12. 2350 m
Reteach
13
13
5
; sec θ =
; cot θ =
12
5
12
1.
3
5
4
5
3
4
8. sin θ =
3
4
3
; cos θ = ; tan θ =
5
5
4
2.
12
13
5
13
12
5
csc θ =
5
5
4
; sec θ = ; cot θ =
3
4
3
3.
8
17
15
17
8
15
csc θ =
9. sin θ =
9
40
9
; cos θ =
; tan θ =
41
41
40
csc θ =
41
41
40
; sec θ =
; cot θ =
9
40
9
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
A30
Holt McDougal Algebra 2
4.
15
17
17
15
2. a. 12
8
17
17
8
c. 15
15
8
8
15
5
5. sin θ =
13
b. 9
13
csc θ =
5
5
12
cot θ =
12
5
Challenge
1. 220 m
2. 454 m
3. 234 m
4. 70 ft
9
3
or
15
5
e.
12
4
or
9
3
3. An isosceles triangle has 2 sides that are
equal in length. Since the hypotenuse is
the longest side of a right triangle, the
equal sides must be the 2 legs. Therefore,
the 2 angles are the same. Since the
angles are equal, both 45°, they have the
same sine.
12
13
cos θ =
sec θ =
13
12
tan θ =
d.
10-2 ANGLES OF ROTATION
Practice A
5. 54 ft
1.
Problem Solving
1. a. a
b. tan 75° =
a
180
c. 672 yd
2. a. 380 yd
b. cos 60° =
380
c
c. 760 yd
3. B
4. H
5. D
6. J
2.
Reading Strategy
1. a. 5
b. 8
c. 9.95
d.
8
9.95
e.
5
8
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
A31
Holt McDougal Algebra 2