Download Trigonometric Ratios

Name
LESSON
8-2
Date
Class
Reteach
Trigonometric Ratios
Trigonometric Ratios
leg opposite !A
4 ! 0.8
sin A ! ______________ ! __
5
hypotenuse
"
hypotenuse
leg opposite !A
leg adjacent to !A
3 ! 0.6
cos A ! ________________ ! __
5
hypotenuse
!
leg opposite !A
4 ! 1.33
tan A ! ________________ ! __
3
leg adjacent to !A
#
leg adjacent to !A
You can use special right triangles to write trigonometric ratios as fractions.
leg opposite !Q
sin 45" ! sin Q ! ______________
hypotenuse
"2
!"#
X qi
$
x ! ___
1
! ____
!
!
x"2
!
"2
! ___
2
5
2
1
!"#
$X
X
X
3
'&#
X
%&#
4
X qi
%
6
!
"2 .
So sin 45" ! ___
2
Write each trigonometric ratio as a fraction and as a decimal
rounded to the nearest hundredth.
1. sin K
2. cos H
15 ! 0.88
___
+
()
*
*
("
(
15 ! 0.88
___
17
17
3. cos K
4. tan H
8 ! 0.47
___
8 ! 0.53
___
17
15
Use a special right triangle to write each trigonometric ratio as a fraction.
5. cos 45"
6. tan 45"
!
"2
___
1!1
__
1
2
7. sin 60"
8. tan 30"
!
!
"3
___
"3
___
2
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
3
14
Holt Geometry
Name
Date
Class
Reteach
LESSON
8-2
Trigonometric Ratios
continued
You can use a calculator to find the value of trigonometric ratios.
cos 38" ! 0.7880107536 or about 0.79
You can use trigonometric ratios to find side lengths of triangles.
Find WY.
adjacent leg
cos W ! __________
hypotenuse
Write a trigonometric ratio that involves WY.
7.5 cm
cos 39° ! ______
WY
7
Substitute the given values.
7.5
WY ! ______
cos 39°
WY ! 9.65 cm
CM
8
—
Solve for WY.
Simplify the expression.
9
Use your calculator to find each trigonometric ratio. Round to the nearest
hundredth.
9. sin 42"
10. cos 89"
0.67
0.02
11. tan 55"
12. sin 6"
1.43
0.10
Find each length. Round to the nearest hundredth.
13. DE
14. FH
$
&
(&+-./
%(#
'
(*+,
$)#
#
%
(
39.65 m
6.01 in.
15. JK
16. US
,
3
4
%!/'+,,
$$/"+0,
+
(*#
''#
*
5
32.91 mm
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
55.32 cm
15
Holt Geometry
/BNF
%BUF
$MBTT
/BNF
1SBDUJDF"
-&440/
*O&YFSDJTFToGJMMJOUIFCMBOLTUPDPNQMFUFFBDI
EFGJOJUJPO5IFOVTFTJEFMFOHUITGSPNUIFGJHVSFUP
DPNQMFUFUIFJOEJDBUFEUSJHPOPNFUSJDSBUJPT
"
C
!
A
#
B
PQQPTJUF
5IFTJOFTJO
PGBOBOHMFJTUIFSBUJPPGUIFMFOHUIPGUIFMFH
IZQPUFOVTF
UIFBOHMFUPUIFMFOHUIPGUIF
5SJHPOPNFUSJD3BUJPT
6TFUIFGJHVSFGPS&YFSDJTFTo8SJUFFBDIUSJHPOPNFUSJD
SBUJPBTBTJNQMJGJFEGSBDUJPOBOEBTBEFDJNBMSPVOEFEUP
UIFOFBSFTUIVOESFEUI
!
TJO"
UBO#
D
C
B
DPT-
-
@@
#$
,
UBO.
@@
r @@@
DPT
^
r @@@
UBO
"
DPT
UBO
^
r @@@
^
r UBO
.
!
6TFBDBMDVMBUPSUPGJOEFBDIUSJHPOPNFUSJDSBUJP3PVOEUPUIF
OFBSFTUIVOESFEUI
@@
^
@@
TJO
6TFUIFGJHVSFGPS&YFSDJTFTo8SJUFFBDIUSJHPOPNFUSJD
SBUJPBTBTJNQMJGJFEGSBDUJPOBOEBTBEFDJNBMSPVOEFEUP
UIFOFBSFTUIVOESFEUI
@@@
6TFTQFDJBMSJHIUUSJBOHMFTUPXSJUFFBDIUSJHPOPNFUSJDSBUJPBTB
TJNQMJGJFEGSBDUJPO
C
UBO# @@@
C
@@@
PQQPTJUF
5IFUBOHFOUUBO
PGBOBOHMFJTUIFSBUJPPGUIFMFOHUIPGUIFMFH
BEKBDFOU
UPUIFBOHMF
UIFBOHMFUPUIFMFOHUIPGUIFMFH
B UBO" @@@
UBO"
@@@
D
DPT"
B
DPT# @@@
DPT" @@@
D
@@@
TJO#
BEKBDFOU
5IFDPTJOFDPT
PGBOBOHMFJTUIFSBUJPPGUIFMFOHUIPGUIFMFH
IZQPUFOVTF
UPUIFBOHMFUPUIFMFOHUIPGUIF
#
@@@
"
DPT#
@@@
C
TJO# @@@
B
TJO" @@@
D
$MBTT
1SBDUJDF#
-&440/
5SJHPOPNFUSJD3BUJPT
TJO-
%BUF
TJO¡
DPT¡
UBO¡
6TFBDBMDVMBUPSUPGJOEFBDIUSJHPOPNFUSJDSBUJP3PVOEUPUIFOFBSFTUIVOESFEUI
DPT
TJO
UBO
'JOEFBDIMFOHUI3PVOEUPUIFOFBSFTUIVOESFEUI
$
"'(
"
2
#
N
8
)%+,
2
4
'-(
5IFHMJEFTMPQFJTUIFQBUIBQMBOFVTFTXIJMFJUJTMBOEJOHPO
BSVOXBZ5IFHMJEFTMPQFVTVBMMZNBLFTBBOHMFXJUIUIF
HSPVOE"QMBOFJTPOUIFHMJEFTMPQFBOEJTNJMFGFFU
GSPNUPVDIEPXO6TFUIFUBOHFOUSBUJPBOEBDBMDVMBUPSUP
GJOE &'UIFQMBOFTBMUJUVEFUPUIFOFBSFTUGPPU
$PQZSJHIUªCZ)PMU3JOFIBSUBOE8JOTUPO
"MMSJHIUTSFTFSWFE
/BNF
$
&
MI
%BUF
$MBTT
)PMU(FPNFUSZ
6%+,
-!(
!
5SJHPOPNFUSJD3BUJPT
Y @@@
@@@@
^
^
Yr r ^
r
@@@
-%01/
DN
#OPYRIGHTÚBY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
1
.)(
X
3
"$(
'X
X
8SJUFFBDIUSJHPOPNFUSJDSBUJPBTBGSBDUJPOBOEBTBEFDJNBM
SPVOEFEUPUIFOFBSFTUIVOESFEUI
#
DPT)
TJO,
z
@@@
X
*$(
4
X qi
*
6
+
#-
!
*
#)
(
z
@@@
DPT,
UBO)
z
@@@
z
@@@
6TFBTQFDJBMSJHIUUSJBOHMFUPXSJUFFBDIUSJHPOPNFUSJDSBUJPBTBGSBDUJPO
DPT
UBO
^
r @@@
@@ TJO
UBO
^
^
r @@@
#)%3&
LN
5
2
.)(
X qi
'
r 4PTJO @@@
-)(
r @@@
"/6%2&
$PQZSJHIUªCZ)PMU3JOFIBSUBOE8JOTUPO
"MMSJHIUTSFTFSWFE
#
^
../#%+,
../#%+,
MFHBEKBDFOUUP"
MFHPQQPTJUF2
TJO TJO2 @@@@@@@@@@@@@@
IZQPUFOVTF
6TFUIFGPSNVMBZPVEFWFMPQFEJO&YFSDJTFUPGJOEUIFNJTTJOHTJEFMFOHUIJO
FBDIUSJBOHMF3PVOEUPUIFOFBSFTUIVOESFEUI
.%3& !)(
!
:PVDBOVTFTQFDJBMSJHIUUSJBOHMFTUPXSJUFUSJHPOPNFUSJDSBUJPTBTGSBDUJPOT
HJWFT C CY Y Z B 4VCTUJUVUJOHC CY D B Y
@
@
#VUDPT" D TPY DDPT""OPUIFSTVCTUJUVUJPOHJWFT
B C D CDDPT"
))(
MFHPQQPTJUF"
MFHPQQPTJUF"
UBO " @@@@@@@@@@@@@@@@ @@ z
MFHBEKBDFOUUP"
JO
"
IZQPUFOVTF
MFHBEKBDFOUUP"
DPT " @@@@@@@@@@@@@@@@ @@ IZQPUFOVTF
*UBMTPTIPXTUIBUC Y
Z B &YQBOEJOHUIFMBUUFSFRVBUJPO
)PMU(FPNFUSZ
3FUFBDI
MFHPQQPTJUF"
TJO " @@@@@@@@@@@@@@ @@ IZQPUFOVTF
#
B
1PTTJCMFBOTXFS5IF1ZUIBHPSFBO5IFPSFNTIPXTUIBUY Z D -/-%2&
$MBTT
5SJHPOPNFUSJD3BUJPT
GU
%BUF
A
C
"
5IFMBXPGDPTJOFTJTBGPSNVMBUPGJOEUIFMFOHUIPGUIFUIJSETJEFPG
BOZUSJBOHMFHJWFOUIFMFOHUITPGUXPTJEFTBOEUIFNFBTVSFPGUIF
C
A
Y
JODMVEFEBOHMF/PUF5IFMBXPGDPTJOFTBQQMJFTUPBOZUSJBOHMF
CVUEFSJWJOHJUGPSBOPCUVTFUSJBOHMFSFRVJSFTNPSFLOPXMFEHFUIBO
!
BX
X
ZPVIBWFMFBSOFETPGBS
*OUIFGJHVSFTVQQPTFUIFMFOHUITCBOED
B
BOEUIFNFBTVSFPG"BSFLOPXO%FWFMPQUIFMBXPGDPTJOFTUPGJOEB
&YQMBJOZPVSBOTXFS)JOU6TFUIFDPTJOFGVODUJPOBOEUIF1ZUIBHPSFBO5IFPSFN
GU
%&
/BNF
-/#%01/
N
ZE
$PQZSJHIUªCZ)PMU3JOFIBSUBOE8JOTUPO
"MMSJHIUTSFTFSWFE
-*(
-/"%&
&'
"
6TFUIFGPSNVMBZPVEFWFMPQFEJO&YFSDJTFUPGJOEUIFBSFBPGFBDIUSJBOHMF
3PVOEUPUIFOFBSFTUIVOESFEUI
.*(
%
)#(
$
LN
45
##%+,
#
'%+,
GFFU
1PTTJCMFBOTXFS%SBXBOBMUJUVEFGSPN#BOEDBMMJUTMFOHUII5IFO
@
TJO " @I
D TPI DTJO"5IFGPSNVMBGPSUIFBSFBPGBUSJBOHMFJT
"SFB @@CBTFIFJHIU4VCTUJUVUJPOHJWFT"SFB@@ CDTJO"
*)(
*#%45
&
2
-&440/
-'(
,
NJ
,.
$
4
)/#%3&
5SJHPOPNFUSJD3BUJPT
-/'%&
DN
%
%
(JWFOUIFMFOHUITPGUXPTJEFTPGBUSJBOHMFBOEUIFNFBTVSFPGUIF
JODMVEFEBOHMFUIFBSFBPGUIFUSJBOHMFDBOCFGPVOE*OUIFGJHVSF
TVQQPTFUIFMFOHUITCBOEDBOEUIFNFBTVSFPG"BSFLOPXO%FWFMPQ
BGPSNVMBGPSGJOEJOHUIFBSFB&YQMBJOZPVSBOTXFS)JOU%SBXBOBMUJUVEF
*"(
)*
'-(
-
—
1SBDUJDF$
-&440/
JO
3
GFFU
34
+
$/!%&0
#$$%2&
)
:
3
0
NN
21
'
))(
.#(
9;
*!(
')%&&
##%&
#%
1
(
#'/'%01/
6TFBDBMDVMBUPSBOEUSJHPOPNFUSJDSBUJPTUPGJOEFBDIMFOHUI3PVOEUPUIF
OFBSFTUIVOESFEUI
9
GU
)PMU(FPNFUSZ
$PQZSJHIUªCZ)PMU3JOFIBSUBOE8JOTUPO
"MMSJHIUTSFTFSWFE
)PMU(FPNFUSZ
(OLT'EOMETRY
Reteach
LESSON
8-2
Challenge
LESSON
Trigonometric Ratios
8-2
continued
You can use a calculator to find the value of trigonometric ratios.
Trigonometry was first used to help build pyramids, but the greatest demand for
trigonometry came from astronomers. To illustrate the trigonometric functions,
astronomers used arcs and chords in circles.
cos 38! ! 0.7880107536 or about 0.79
You can use trigonometric ratios to find side lengths of triangles.
Follow the steps to explore trigonometric functions in a circle. You will need
a compass, a protractor, a ruler, and a calculator. Record your measurements
in the table. Hint: The trigonometric ratio cosine has a reciprocal ratio called
1
1 .
secant, defined as secant T ! ________
. In abbreviated form, sec T ! _____
cosine T
cos T
Find WY.
adjacent leg
cos W " __________
hypotenuse
Write a trigonometric ratio that involves WY.
7.5 cm
cos 39° " ______
WY
7
Substitute the given values.
7.5
WY " ______
CM
8
—
1. With a compass, draw a circle of radius 1 inch, centered at T.
Draw an acute angle, !RTU, in which R and U are points on the circle.
Solve for WY.
cos 39°
WY ! 9.65 cm
Simplify the expression.
2. Draw a line tangent to the circle at U.
_
3. Extend radius TR until it intersects the line that is tangent to the circle at U.
Label the point of
intersection S.
_
_
4. Extend TU and TS so you can make the following measurements accurately.
With a protractor measure !T. Given TU " 1 inch, measure TS and US in inches.
5. Use your calculator to find tan T. To find sec T, find cos T. Then find the
reciprocal by using the x#1 key. This will give sec T.
9
Use your calculator to find each trigonometric ratio. Round to the nearest
hundredth.
9. sin 42!
Trigonometric Ratios
10. cos 89!
0.67
0.02
11. tan 55!
3
12. sin 6!
1.43
0.10
2
Find each length. Round to the nearest hundredth.
13. DE
5
4
14. FH
$
!(#)*+
&
'
,!'
!"#$
%&'
#
%
(
39.65 m
15. JK
Possible
answer: 63"
16. US
,
3
4
%%+/#0$
!"'
..'
*
5
32.91 mm
15
LESSON
8-2
Holt Geometry
LESSON
8-2
-
+
4.80 ft
9.49 cm
4. The hypotenuse of a right triangle
measures 9 inches, and one of the acute
angles measures 36°. What is the area of
the triangle? Round to the nearest square
inch.
61.4 cm
Reading Strategies
Opposite
Sine " __________
Hypotenuse
Can
Attract
Horses
Tigers
Or
Alligators
19 in2
Adjacent
Cosine " __________
Hypotenuse
Opposite
Tangent " ________
Adjacent
!
24 " ___
12
Example: sin B " ___
26
13
10 " ___
5
Example: cos B " ___
26
13
#
"
24 " ___
12
Example: tan B " ___
5
10
1. Make up your own mnemonic for remembering the ratios.
S
Answers will vary.
O
Choose the best answer.
5. A 14-foot ladder makes a 62° angle with
the ground. To the nearest foot, how far
up the house does the ladder reach?
A 6 ft
H
6. To the nearest inch, what is the length of
the springboard shown below?
C
A
H
B 7 ft
C 12 ft
D 16 ft
IN
—
7. What is EF,
the measure of
the longest side
of the sail on the
model? Round to
the nearest inch.
A 31 in.
B 35 in.
C 40 in.
D 60 in.
Holt Geometry
Use a Mnemonic
Such
Outrageous
Hats
CM
3. A right triangle has an angle that
measures 55°. The leg adjacent to this
angle has a length of 43 cm. What is the
length of the other leg of the triangle?
Round to the nearest tenth.
16
You can remember the trigonometric ratios by using a mnemonic:
one word for the first letter of each word in the equations. Here is
a sample mnemonic and examples for the triangle at right.
2. Find the perimeter of the triangle. Round
to the nearest hundredth.
—
secT
Possible
answer: 2.20
_
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Trigonometric Ratios
,
length TS
The length of US is close in value to tan T, and the length of TS is close in
value to sec T.
Problem Solving
1. A ramp is used to load a 4-wheeler onto a
truck bed that is 3 feet above the ground.
The angle that the ramp makes with the
ground is 32°. What is the horizontal
distance covered by the ramp? Round to
the nearest hundredth.
tanT
Possible
Possible
15
answer: 1.96 answer: 1 ___
inches 16
_
55.32 cm
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Possible
answer: 1 _3_
inches 4
The tangent of an angle_
is defined as opposite over adjacent. The
radius of 1 inch.
adjacent side of !T is TU, which in your diagram is the_
6. Do you see
_any connection between the length of US and tan T and/or the
length of TS and sec T ? Possible answer:
,-+.#$$
+
length US
Measure of !T
6.01 in.
F 24 in.
G 36 in.
&
O
A
2. How are the ratios for sine and cosine alike?
8. Right triangle ABC is graphed on the
coordinate plane and has vertices at
A(#1, 3), B(0, 5), and C(4, 3). Which is
a true trigonometric ratio for the measure
of !C?
F tan C " _1_
2
IN
% —
T
H 38 in.
J 127 in.
'
Both ratios have Hypotenuse in the denominator.
3. Which side is not used when finding the tangent of an angle?
hypotenuse
4. Use the triangle above to find the following. Write the ratios both as a fraction
and as a decimal rounded to the nearest hundredth.
G tan C " 2
a. sin A "
!
"5
H sin C " ___
10
___
26
b. cos A "
0.38
3
24
___
26
0.92
c. tan A "
10
___
24
0.42
J cos C " 2
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
17
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
001-062_Go07an_CRB_c08.indd 54
Process Black
Holt Geometry
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
54
18
Holt Geometry
Holt Geometry
5/11/06 5:48:29 PM