Trigonometry/ProOneSoftware-Trigonometry-Win31.iso/trig/chapter1.2r

Text File | 1995-04-09 | 8KB | 294 lines

 292
à 1.2ïTrigonometric Ratios and Inverses

äïPlease find the following trigonometric ratios of the
êêgiven acute angle.

âêêïFind the trigonometric ratios of the 30° angle.

êêêê sin 30°ï=ï1/2êcsc 30°ï=ï2
êêêê cos 30°ï=ï√3/2ë sec 30°ï=ï2/√3
êêêê tan 30°ï=ï1/√3ë cot 30°ï=ï√3

@fig1201.bmp,10,100
éSïIn the last section, we looked at solving special triangles by
remembering the common proportions of the 30°-60° triangle and the 45°-
45° triangle.ïIn general, however, right triangles that are not one of
ç two triangles, require different methods to find the lengths of
the sides and the measures of the angles.ïWe will see in the next sec-
tion that in order to solve an arbitrary right triangle, given two sides
or one side and an acute angle, we will need to find trigonometric ra-
tios and/or use inverses.
è The trigonometric ratios of an acute angle are just the ratios of
two sides of the triangle.ïThey are given below where "opp" means the
side opposite the angle, "adj" means the side adjacent to the angle, and
"hyp" means the hypotenuse of the triangle.
êêè oppêêèadjêêè opp
#êsine Θï=ï───,ècosine Θï=ï───,ètangent Θï=ï───
êêè hypêêèhypêêè adj
êêê hypêêèhypêêë adj
#êcosecant Θï=ï───,èsecant Θï=ï───,ècotangent Θï=ï───
êêê oppêêèadjêêë opp
These ratios will be abbreviated as sinΘ, cosΘ, tanΘ, cscΘ, secΘ, and
cotΘ.ïNext to the figure below you will see the list of six ratios for
both angle "Θ" and angle "φ".
êêêsin Θï=ïa/cêïcos φï=ïa/c
êêêcos Θï=ïb/cêïsin φï=ïb/c
êêêtan Θï=ïa/bêïcot φï=ïa/b
êêêcsc Θï=ïc/aêïsec φï=ïc/a
êêêsec Θï=ïc/bêïcsc φï=ïc/b
êêêcot Θï=ïb/aêïtan φï=ïa/b
êêêYou can see why it is better to remember the
@fig1202.bmp,10,135
êêêratios in terms of opposite, adjacent, and
êêêhypotenuse.ïFor example, sin φ = opp/hyp = b/c.
êêêè There are some patterns, however, that can
êêêbe observed about trigonometric ratios.ïThe most
êêêimportant one is that the six ratios for a
êêêfixed angle have the same values as the
êêêcorresponding ratios for all other triangles
êêêwith this angle.ïThis is true because of similar
êêêtriangles, and it will allow us to use the cal-
êêêculator or a trigonometric table to find a
êêêparticular ratio for a given angle.
Another pattern can be seen in the above list of ratios.ïThe sinΘ is
the reciprocal of the cscΘ, the cosΘ is the reciprocal of the secΘ, and
tanΘ is the reciprocal of the cotΘ.ïThis pattern is called the
reciprocal relationship.
è Another pattern involves complementary angles (angles that add up to
90°).ïIn the figure, angles A and B are complementary.ïYou can see
that sinΘ = cosφ, tanΘ = cotφ, and secΘ = cscφ.ïIn general, any trigo-
nometric function of an acute angle equals the corresponding cofunction
of its complement (the prefix "co" comes from the word complementary).
 1êêFind the sin 45°.
#êêêë ┌─
#êêêïA)ïá2êêëB)ï1

êêêïC)ï1/√2êêèD)ïå of ç

@fig1203.bmp,15,118
ü

êêêêêoppê 1
#êêêïsin 45°ï=ï───è=è──
êêêêêhypê√2
Ç C
 2êêFind the cos 60°.

êêêïA)ï2êêë B)ï1/2

êêêïC)ï√3/2êêèD)ïå of ç

@fig1204.bmp,15,118
ü

êêêêêadjê1
#êêêïcos 60°ï=ï───è=è─
êêêêêhypê2
Ç B
 3êêFind the tan 30°.

êêêïA)ï1/√3êêèB)ï2

êêêïC)ï1/2êêè D)ïå of ç

@fig1205.bmp,15,118
ü

êêêêêoppê 2êï1
#êêêïtan 30°ï=ï───è=è───è=è──
êêêêêadjê2√3ê√3
Ç A
 4êêFind the sec 45°.

êêêïA)ï√2êêëB)ï1/√2

êêêïC)ï1êêë D)ïå of ç

@fig1206.bmp,15,118
ü

êêêêêhypê3√2
#êêêïsec 45°ï=ï───è=è───è=è√2
êêêêêadjê 3
Ç A
 5êêFind the cot 60°.

êêêïA)ï2êêë B)ï√3

êêêïC)ï1/√3êêèD)ïå of ç

@fig1207.bmp,15,118
ü

êêêêêadjê 4êï1
#êêêïcot 60°ï=ï───è=è───è=è──
êêêêêoppê4√3ê√3
Ç C
 6êêFind the csc 30°.

êêêïA)ï1/2êêè B)ï2

êêêïC)ï√3êêëD)ïå of ç

@fig1208.bmp,15,118
ü

êêêêêhypê2
#êêêïcsc 30°ï=ï───è=è─è=è2
êêêêêoppê1
Ç B
 7êè Use your calculator to find the sin 49.3°.

êêêïA)ï.478êêèB)ï.758

êêêïC)ï.231êêèD)ïå of ç

@fig1209.bmp,15,118
üïTo use your built-in calculator to find the sin 49.3°, you
should first click on the "deg" button.ïNext, you should enter 49.3,
and finally click on the "sin" button.ïYou will see that the sin 49.3°
is approximately .758.ïNotice that it is not necessary to know the
sides to find the sin 49.3° when you use the calculator or a trigonome-
tric table.ïThat is because the sin 49.3° is approximately .758 for all
right triangles with this angle no matter how big the triangle.ïIn
effect, the calculator just gives you the correct ratio from a prere-
corded list of ratios that have been calculated for each angle.
Ç B
 8ê Use your calculator to find the tan 23.7°.

êêè A)ï.439êêèB)ï.215

êêè C)ï.931êêèD)ïå of ç


ü


êêê tan 23.7°ï=è.439
Ç A
 9ê Use your calculator to find the cos π/8.

êêè A)ï.139êêèB)ï.572

êêè C)ï.924êêèD)ïå of ç


üïTo use your calculator to find the con π/8, you should first
click on the "rad" button.ïNext, you should enter "π" divided by "8",
and finally you should click on the "cos" button.ïYou will see that
the cos π/8 is approximately .924.
Ç C
 10êUse your calculator to find the csc 72.45°.

êêè A)ï2.631êêïB)ï4.712

êêè C)ï1.049êêïD)ïå of ç

üïSince you have no "csc" button, it is necessary to use the
reciprocal relationship.ïThe csc 72.45° is equal to the reciprocal
of the sin 72.45°.ïYou should click on the "deg" button, enter 72.45,
click on the "sin" button, and then click on the "1/x" button.ïYou
should see approximately 1.049.
Ç C
äïPlease find the inverse trigonometric functions of the
êêgiven ratios.
âêFind the following inverse trigonometric values.

#êêêë sinúî .5è=è30°
#êêêë tanúî 1è=è45°
#êêêë secúî 2è=è60°
éSïIn the next section we will need to use the inverse trigonome-
tric functions to find an acute angle of a triangle given two sides of
the triangle.ïFor example, if you know that the ratio of the opposite
side over the hypotenuse is 1/2, is it possible to find the angle?ïThe
answer is yes if you recognize that you have one of the special trian-
gles or if you use the inverse trigonometric buttons on the calculator.
@fig1210.bmp,15,118
êêêIn the figure, you can see that ç are the sides
êêêof a 30°-60° triangle, so the angle at A must
êêêbe 30°.ïTo use your calculator, you would recall
êêêthat the ratio, opposite over the hypotenuse, in-
êêêvolves the "sin" function.ïFirst, click on the
êêê"deg" button, enter "1/2", then click on the
#êêê"sinúî" button.ïYou should see 30°.ïIf your cal-
êêêculator is in the "rad" mode, your answer
êêêwill be given in radians, approximately .524.
#êêêTo find tanúî 1, you could recognize that
êêêthis involves the 45°-45° triangle, or you could
êêêuse your calculator. Click on the "deg" button,
#enter "1", and click on the "tanúî" button.ïYou should see 45°.
#è To find the secúî 2 using your calculator, you must use the recipro-
cal relationship.ïFirst, click on the "deg" button, enter "2", click on
#the "1/x" button, and finally click on the "cosúî" button.ïYou should
see approximately 60°.ïWe will look at the inverse trigonometric func-
tions in more detail a little later.
 11êêêêè√3
#êêêëFind the cosúî ──.
êêêêêê2

êêëA)ï60°êêèB)ï30°

êêëC)ï45°êêèD)ïå of ç
ü

êêêêï√3
#êêêècosúî ──è=è30°
êêêêè2
Ç B
# 12êêêêè┌─
#êêêëFind the secúî á2.


êêëA)ï30°êêèB)ï60°

êêëC)ï45°êêèD)ïå of ç
ü

#êêêêï┌─
#êêêèsecúî á2è=è45°

Ç C
 13
#êêêëFind the tanúî 23.5


êêëA)ï87.56°êêB)ï46.2°

êêëC)ï1.235°êêD)ïå of ç
ü


#êêêètanúî 23.5è=è87.56°

Ç A
 14
#êêëFind the sinúî .86 in radian measure.


êêëA)ï30°êêèB)ï2.614

êêëC)ï1.035êê D)ïå of ç
ü
êêë First, click on the "rad" button.

#êêêèsinúî .86è≈è1.035

Ç C
 15
#êêëFind the cscúî 1.7 in radian measure.


êêëA)ï.731êêïB)ï.824

êêëC)ï.629êêïD)ïå of ç
ü
êêë First, click on the "rad" button.

#êêè cscúî 1.7è=èsinúî (1/1.7)è≈è.629

Ç C