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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