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and sides. Also, find the general expression for the length of the base in terms of altitude and angles at base.

9. From the top of a tower 108 feet high the angles of depression of the top and bottom of another tower standing on the same horizontal plane are found to be 28° 56′ and 53° 41′ respectively. Find the distance between the towers, the height of the second tower, and the distance between the summits of the two towers.

10. In a circle of radius r, express the length of each side, and of the apothegm, of a regular inscribed polygon of n sides. Find first the special values for the triangle, square, pentagon, hexagon, and octagon. Ans. For the octagon: side = 2r sin 221°; apothegm = r cos 2210.

11. If the side of a regular octagon is 10 metres, what are the radii of the inscribed and circumscribed circles?

12. At what altitude is the sun when a tower 20 metres high casts a shadow 75 metres long upon a horizontal plane?

13. It was found that the length of the shadow of a monument upon a horizontal plane diminished 22 metres when the sun's altitude increased from 30° to 45°. What was the height of the monument?

14. If ẞ is one of the acute angles of a right triangle, and e its hypothenuse, express the altitude in terms of c and ß.

15. Two lighthouses, each 30 metres above the sea and 500 metres apart, are seen by a ship in line with them to differ 1° in elevation above the horizon. What is the distance from the

nearer, supposing the ocean a plane?

16. The great pyramid of Gizeh is 762 feet square at its base, and each side makes an angle of 51° 51' with the horizon. Find

(a) Its height if continued to its apex.

(b) The slope of its edges.

Actually, instead of being continued to its apex, it terminates in a platform 32 feet square. Find

(c) The perpendicular height of this platform above the base. (d) The length of each edge from the corner of the platform to the corner of the base.

CHAPTER IV.

RELATIONS BETWEEN FUNCTIONS OF SEVERAL ANGLES.

The Addition and Subtraction Theorems.

40. PROBLEM. To express the sine and cosine of the sum of two angles in terms of the sines and cosines

of the angles themselves.

Solution. Let XOM = a and MON = ß be the two angles. XON = a + ẞ is then their sum.

Let ON be the unit radius. From N drop

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N

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Then

sin (a+6)= NQ= NR+MP;

cos (a+B) OQ = OP MR.

=

OQS and NMS being right angles, we have

Angle RNM = comp. RSM = comp. OSQ = SOQ =

By § 35,

NM = sin ẞ;

ß;

NR NM cos a = sin ẞ cos a;

OM = cos ẞ;

= α.

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The formulæ (2) and (3) constitute the addition theorem of trigonometry.

To find the corresponding subtraction theorem it is only necessary to change the sign of ß. We have

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Therefore, changing ẞ to ẞ in (2) and (3), we find

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It is, however, interesting to show how these equations may be obtained independently by a geometrical construction.

Let POM be the angle a, and NOM

the angle 6. Then

ΡΟΝ = α - β.
PON

Take ON for the unit radius and

M

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Then

sin (a)

=

=

PN= QR = MQ MR;

cos (ap) OP = OQ+RN.

Because OMN is a right angle,

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P

RN MN sin NMR

Making these substitutions, we have the results (4) and (5) for sin (a) and cos (a — ß).

41. Sine and cosine of twice an angle. If we suppose B = a, we have from (2) and (3) expressions for the sine and cosine of the double of an angle, namely:

sin 2 α = sin a cos a + sin a cos a,

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(6)

2 sin a cos a;

(cos a + sin a) (cos asin a). (7)

sin' a =

Also, by putting for cos' a its value, 1 — sin' a, and vice versa, we

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sin 90° = sin 60° cos 30° + cos 60° sin 30° = 1.

It is required to test this equation by substituting the numerical values of the sines and cosines of 30°, 60°, and 90° (§30), and to test in the same way the equations obtained by putting a = 60° and 6 30° in (3), (4) and (5).

2. Because a α-x+x, we have, from (2),

=

sin a = sin (a

x) cos x + cos (α — x) sin x.

It is required to write the corresponding equations obtained by making the same substitution in (3), and the equations obtained in the same way from (4) and (5) by the identical equation

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3. Derive the addition theorem for the cosine from that for the sine by substituting y 90° or 90° y for in the equation

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(2), and applying the equations (18) of § 29 (Chap. II.).

4. By means of the addition theorem prove the equations sin (a+B+y)= sin a cos ẞ cos y+cos a sin ẞ cos y +cos a cos ẞ sin y.

cos (a+B+y)= cos a cos ẞ cos y

= cos a cos ẞ cos y

- sin a cos ẞ sin y

--

--

NOTE. This is readily done by putting aẞ equations (2) and (3), thus giving

sin a sin ẞ sin y;

sin a sin ẞ cos y

cos a sin ẞ sin y.

for a, and y for ß, in the

sin (a + B+ y) = sin (a + B) cos y + cos (a + ẞ) sin y;
cos (a+B+ y) = cos (a + ß) cos y sin (a +ẞ) sin y;

and then developing by the addition theorem.

into

5. Prove the following values of sin 3a and cos 3a:
sin 3a 3 sin a cos2 α
- sin3 a;
cos 3a cos' a
=

6. Transform the expression

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3 sin' a cos a.

a cos ( a + x) + b cos (ß + x) + c cos (y + x)

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prove that we must also have

a sin (a + x)+b sin (ß + x) + c sin (y + x) = 0,

whatever be the value of x.

42. The Addition Theorem for tangents. Dividing equation (2) by (3), we have

sin (a+B)

=tan (a + B)

cos (a+B)

=

sin a cos

+ cos a sin B

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Dividing both numerator and denominator of the last member by

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We obtain in the same way from (4) and (5),

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EXERCISES.

1. Assuming = 180°, prove by (8) of this section that tan (a+180°) = tan a.

=

2. Assuming a 30°, substitute in (10) the value of tan 30° given in § 30, and thus obtain the value of tan 60°.

43. Products of sines and cosines. Taking the sum and difference of equations (2) and (4) of § 40, and reversing the members of the equation, we find

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