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SYLLABUS, ,'.J
OF A'0
COURSE IN· PLANE TRIGONOMETRY.
r
BOSTON, ,u.S.A. :
GINN &. C01\fPAl.'TY, PUBLISHERS.
181>3.
SYI,LABUS0'" A
COURSE IN PLANE TRIGONOMETRY.
1. Explain briefly tho object of Trigonometry. Show why It
new way of measuriug angles is dosirnbl«. The ratio between
two sides of a right triangle formed hy dl'Opping It porpcndiculnrfrom a point ill one s�de of an angle IIpon the other side may beused to represent the angle. That is if either angl or ratio is
given the other will be determined.
2. Define the sine, cosine, tangent cotangent, secant, nnd
cosecant of an angle as ratios between the sides of n righttriangle containing the angle. These ratios arc called the
Trigonometric Functions of the angle.
sin A =a
= cosBc
cos A =b
= sin B,c
[1JtnnA=�=ctnB,
b\b
ctnA= = tnnBa
csec A =
b= ('ScB,
cscA=C
= secB..
a
-
',;--:,: 2
3> E��kijikh�the formulas";'
· [2}r:�:� :�:�:�:-
l tanA ctnA;; L
sinA_t A--_ an .
cos.zl[3J
[4J
[5J
[6J
4. Show that when one function of"
an angle is given all the
rest can be obtained from it by the aid of the formulas in § 3.
5. O�ain sin 30° and tan 45°. from geometrical figures.Compute the. other functions of 30° and 45°.
Write all the functions of 60°.
6. Solution. of RI;ght, Triangles. If any two parts in addition
to'the right angle are given, provided that one is a side, the
triangle can be completely solved, using only the formulas [1 Jand the relation
In solving, obtain each part separately from the two givenparts. Work examples, using a three-place or four-place table
of natural sines, cosines, etc.
7. In the previous sections of this syllabus the work done
has 3fPl?Jied only to angles less' than 90°. With a slight exten ..
:.;, �(. ��\;'-�sion of 'the original definitions all the results already obtained
t_
'
_•
can be made to apply to angles of any magnitude,
'_:, •,
'_,_, :' __:-_, ,�-.<,_._ , :",_
,'::' f"<',.'/�'��:':��,'.{_;::. "
:.. �,-
- An angle may always be regaided as' £orme�.by:;r-Q�atiirg'·oR�s.ii:le about the vertex, _
from coincidence with theother side to
Wffii1al position, and its magnitude will, depend upon the amount
o.{-this l;otatio�l. Initial andleihliiial sides �f ��n "�hgi�. .
.
Distinguish betw�en positive and negativerotation..Describe the classification of angles by quadrants. ,Draw
angles ill each of the four quadrants.
8. Define the triangle of reference for "anyangle , and draw
it for an angle in each of thefour quadrants. Explain what is
meant by the normal position of an angle. Give the rule for
th� signs of the sides of the triangle of reference.
9. Adopting the definitions [1], show that formulas [2] to
[6J inclusive hold for an angle in any quadrant.
10. The quadrantal angles 0°, 90°, 180°, 270°, 360°, need
special treatment. The's are not strictly covered by the defi
nitions [1], because 110 triangle of reference can be drawn for
them.,
'
There is an advantage in, having values to' represent them;and these values, to be - of rise, should satisfy all the relations
that the functions of other angles satisfy.', Such values can
probably be obtained for ,any quadrantal angle by taking the
limiting values approached 1)y the functions of 'an angle as the
angle approaches the quadrantal- angle in question.Find such values for sine, cosine, tangent, etc., of 0°, 90°,
180°, 2,70°, 360°, and prove that they obey formulas [2j,.',to [6],j ....
mid can therefore be safely used as the functions of these
angles.'
4
Formulas [2] to [6J now apply to all angles without exception.
[7J
I sin I cos I tan I ctn I sec I esc
0° I 0 I 1 I 0 I 00 I 1 I 00
fJO° I 1 I 0 I 00 I· 0 I 00 I 1
iso- I 0 1-1 I 0 I 00 1-1 I 00
270° 1-1 I 0 1 00 I 0 I 00 1-1360° I o I 1 1 0 I 00 I 1 I' 00
11. Graphicctl represenuuion. of the functions of em omqle.Show that if a circle with the radius unity is described from the
vertex of an acute angle as a centre, a set of lines can be drawn
whose lengths are the values of the functions of the angle.Give a brief description of each of these lines.
Show that lines drawn according to 'this description will rep
resent in sign as well as iil uiagnitude thc functions of an anglein m.ly qundraut and also the functions of the quadrantalangles.
12. Show that a trigonometric table gi.ving the functions 01
angles in the first quudruut is a complete table, i.e., that the
functions of any angle can be obtained from those of some acute
angle. An angle in the second quadrant may be taken as
90° + cp or as 180°- cp, of the third as 180°+ cp or as 2700-cp,of the fourth as 270° + cp or .as 360° - cp, where cp is acute.
By the aid of the unit circle obtain formulas for the functions
of �)O0 - cp, 90° + cp, 180°- cp, 180°+ ¢, 270°- ¢, 270°+ cp,
3600-cp, -cpo Show that these formulas hold, no matter what
the magnitude of cpo
5 •••: �."i·: i"·:.· � ••:.: t· :..... ,
Prove geometrically the Iormulus :
[KJ sin (a + (3) = sin a cosf3 + cos a sin p,[!JJ sin (a - (3) = sin a cosf3 - cos a sinf3,
[10J co�(a + /1) = coau 'osf3 - sin a sinf3,
[11 J co '(a - (3) -= cos« cosf3 + sin a sinf3,
Ill-awing the figures for the case where a f3, 0.+ f3, 0.- f3, are
all acute. Show that essentially the same proof holds when the
angles considered are of any magnitude,Show that by the aid of formulas [�J to [11 J the sines and
cosines of any sum of angles may be obtained.
14. From § 13 obtain the formulas :
[12J tan(a+f3)= tantL+tnn,B.1 - tan a tan f3
[13J tall (a _ (3) =tan a - tan f3
.
1 + tana tanf3
15. Establish the formulas:
[14J
[15J
[16J
sin 2 a= 2 sin a cos a.
cos2a= Cos2a-sinl!a =1-2sin2a = 2 C082 (.I, -1.
tan2a= 2tana.
1- tan2a
[17J sin � = vi (1 - cos«).
cos� = VHl + cosa).
tan � =11 - coso..
2 'J 1 + coso.
[18J
[19]
Show that formulas for sine and cosine of 3 a, 4 a, etc., can
be readily deduced.
16. P1'Ov� that in any triangle the sides are to each other as
the sines of the opposite angles,
[20J Si:A = Si� B= Si� 0.
17. Prove that in any triangle the square on one side is
the slim of the squures 011 the other two sides minus twice the
product. of these sides into tile cosine of the angle betweenthem.
[21J a2 = b2 + c? - 2 be cosA.
18. Classify the cases that can arise in the solution of
oblique triangles.
Case 1. Given two angles and a side.
Case 2. Given two sides and an angle not included bythem.
Case 3. Given two sides and the included angle.Case 4. Given three sides.
Show that Case 1 can be easily treated by § 16. Examples.
19. Show that Case 2 can be solved by § 16, but that
there will generally be. two solutions. Consider special cases.
Examples.
20. Show that Case 3 and Case 4 can be solved by § 17.
Examples.
21. Explain the theory of Logarithms and the use of Logarithmie 'I'ables.
'l
22. Sbow that when the work of solving triauglO;,s is done
by the aid of logarithms, some of the methods described above
need modifleation.
Consider, in the solution of Right Triangles, the case where
two sides are given, or the hypothenuse and a side.
[22J a=:v(c- b)(c+ b).
23. Prove, by the aid of §§ 16 and 13, that the sum of two
sides of a triangle is to their difference as the tangent Of' one
half the snm of the opposite angles is to the tangent of one-half
their difference.
[23J �=tanHA+B>,a- b tanHA -B)
Show how § 18, Case 3 can be treated by the aid of this
theorem. Examples.
24. Throw [21J into a convenient form for use with logarithms in solving § 18, Case 4.
If s=i(a+b+c),
[24]A �s(s-a)cos-= .
2 be
. A_�(S-b)(S--e)Sln -- •
2 be
tan� = /(.'f - b}(s - ct:2 '\J s(s-a)
[25J
[26J
Formulas for cos�, sin.Q, ta'n�, can be obtained from [24],2 2 2
[25], and [26], by changing a into b b into c, and c into C£ j
for cos Q, sinQ, and tan Q, by changing a into c, b into a, and222
e into b.
8
[26J is the most convenient formula when the triangle is to
be solved completely. Examples.
25·. Miscellaneous examples. Heights and Distances.
26. A?·ew1. Obtain a formula for the area of a triangle in
terms of two sides and the included angle.
[27J ]{= �(lb sin O.
Examples.
W. E. BYERLY,
Professor 0/ Muthemutic« "in Harvard University.
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Syallbus of a course in plane trigon
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