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Exercise 84. Review Problems

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1. The angle of elevation of the top of a vertical cliff at a point 575 ft. from the foot is 32° 15'. Find the height of the cliff.

2. An aeroplane is above a straight road on which are two observers 1640 ft. apart. At a given signal the observers take the angles of elevation of the aeroplane, finding them to be 58o and 63o respectively. Find the height of the aeroplane and its distance from each observer. 3. Prove that (va CSC x + cotx

cotx)' = 2 (csc x – 1). 4. Given sin x = 2 m/(m2 + 1) and sin y= 2 n/(m2 + 1), find the value of tan (x + y).

5. Find the least value of cos” x + secʻx. 6. Prove that 1-sina x /sin'y = cos x (1 - tan/tan’y). 7. Prove this formula, due to Euler : tan-} + tan-}= . 8. Prove that cot - x cotx = CSC X. 9. Prove that (sin x+i cos x)"=cos n (T-)+ i sin n (---2). 10. Show that log i = į Ti and that log (-i)=- tri.

11. Through the excenters of a triangle ABC lines are drawn parallel to the three sides, thus forming another triangle A'B'C'. Prove that the perimeter of AA'B'C' is 4 r сot 4 A cot } B cott C, wherer

the radius of the circumcircle. 12. Given two sides and the included angle of a triangle, find the perpendicular drawn to the third side from the opposite vertex.

13. To find the height of a mountain a north-and-south base line is taken 1000 yd. long. From one end of this line the summit bears N. 80° E., and has an angle of elevation of 13° 14'; from the other end it bears N. 43° 30' E. Find the height of the mountain.

14. The angle of elevation of a wireless telegraph tower is observed from a point on the horizontal plain on which it stands. At a point a feet nearer, the angle of elevation is the complement of the former. At a point b feet nearer still, the angle of elevation is double the first. Show that the height of the tower is [(a + b)2 – į a?j?.

Prove the following formulas :
15. 2 cos x = cos 2 x +1. 17. 8 cos*x = cos 4x + 4cos 2 x + 3.
16. 2 sina =-

cos 2 x + 1. 18. 4cos8 x = cos 3x + 3 cos x.
19. 4 sinox =- sin 3x + 3 sin x.
20. 8 sin4 x = COS 4 x 4 cos 2 x + 3.

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.

=

tan •

coto sin $.

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RELATIONS OF FUNCTIONS ($$ 13, 14, 89)
1
1

cos .
7. sin
12. coto

17. sin $
CSC.
1

1
8. Cos 0
13. sec

18. tan •
sec
COS •

COS •
1
1

cos
9. tan •
V 14. Csc

19. cot • coto

: 1. 10. sin $ csco=1. 15. tan $ coto 20. 1+tan p=sec* p. 11. cos sec =1. 16. sin’d+cosop=1. 21. 1 + cot d=cscd.

sin •

sin φ

FUNCTIONS OF x y (S$ 90-100)

. 22. sin(x + y)= sin x cos y + cos x sin y. 23. sin (x y)= sin x cos y

cos x sin y. 24. cos (wc + y)= cos x cos y

sin x sin y. 25. cos (x - y) = cos x cos y + sin x sin y.

tan x + tany,

cotx coty - 1 26. tan (x+y)=

28. cot (x + y) = 1 tan x tany

cot y + cota

cotx coty +1 27. tan (x - y)

29. cot (x - y) 1+ tan x tany

cot y — cotx

tan x

tan y

29.

FUNCTIONS OF TWICE AN ANGLE (101) 30. sin 24 2 sin $ cos .

32. cos 2$ = cos – sind. 2 tan •

coto - 1 31. tan 2

33. cot 2$ 1- tan

2 cot •

FUNCTIONS OF HALF AN ANGLE ($ 102)

1 1 coso

coso. 34. sin } $ =V

36. tan } $=V 2

1 + cos 0 1 + cos $

1 + cos 0 35. cos } $ = =

37. cot } € 2.

ços

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38. sin x = 2 sin

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FUNCTIONS INVOLVING HALF ANGLES ($ 101)
- 2 sin cost 40. cos x = cosant-

cosas - sino
2 tane

coto - 1 1 – tan?

2 cot

39. tan x =

41. cot x =

2

SUMS AND DIFFERENCES OF FUNCTIONS (§ 103) 42. sin A + sin B = 2 sin (A + B) cos } (A - B). 43. sin A sin B 2 cos } (A + B) sin }(A B). 44. cos A + cos B = 2 cos }(A + B) cos } (A B). 45. cos A - cos B=- 2 sin }(A + B) sin f(A -- B). sin A + sin B

tan }(A + B). 46.

sin A - sin B tan (A - B)

LAWS OF SINES, COSINES, AND TANGENTS (S$ 105, 111, 112)

sin A 47. Law of sines,

b sin B

a

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FORMULAS IN TERMS OF SIDES (S$ 115, 116) a+b+c

(s a)(s b)(s —c) 50.

53.
2
- b)(3 —c).

(8-6)(s –0). 51. sin ţA

54. tanA= bc

8 (8-a)

= S.

=r

S

S

SS

a).

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52. cos & A :

55. tan A

bc

S
- a

AREAS OF TRIANGLES ($ 118) 56. Area of triangle ABC = 1 ac sin B= fr(a + b + c)= rs =

abc a? sin B sinc Vs(8 - a) (8 -- b) (8 - 0) =

4R 2 sin (B+C)

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INDEX

.

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77, 92

87

.

PAGE

PAGE
Abscissa
78 Depression, angle of

18
Addition formulas

97, 101
Difference of two angles .

100
Algebra, applications to

173
of two functions

105
Ambiguous case
112 Division by logarithms

42, 52
Angle, functions of an

3, 4
of depression
18 Elevation, angle of .

18
of elevation
18 Eliminant

171
negative

Equation

163, 166, 169, 173
positive
77 Euler

181
Angles, difference of
100 Euler's Formula .

181
differing by 90°
92 Expansion in series

180
greater than 360°
Exponential equation .

58
having the same functions 154, 155

series

179
how measured

2
sum of .
97 Formulas, important

185
Antilogarithm .

48
Fractional exponent

57
Areas

66, 128, 141, 142
Functions as lines

23
changes in the

25
Base
40 graphs of

158
Briggs.
39 inverse.

156
line values of

85
Changes in the functions

25
logarithms of

60
Characteristic .

43
of a negative angle

92
negative

44, 51
of an angle

3, 10
Circle

144
of any angle.

82
Circular measure

151 of half an angle
Cologarithm

54
of small angles.

153
Compass

146

of the difference of two angles 100
Complementary angles

7

of the sum of two angles . 97
Conversion table .

30
of 30°, 45°, 60°.

8
Coördinates.

78
of twice an angle .

103
Cosecant

4, 22
reciprocal.

12
Cosine .

4, 16, 116, 180
relations of

12, 13
Cosines, law of

116

variations in.
Cotangent

4, 20
Course .

145
Graphs of functions

158
Coversed sine.

171
Half angles .

104, 123
Decimal table .

30
De Moivre's Theorem
174 Identity

163
Departure

145
Interpolation

31, 32, 48

.

104, 123

.

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86

.

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Parallel sailing
Plane sailing

trigonometry
Polygon, regular .
Positive angle
Power, logarithm of
Practical use of the cosecant

148
145

Tables explained 10, 28, 30, 46, 48, 61
Tangent

4, 18
Tangents, law of

118
Traverse sailing

150
Trigonometric equation

163
identity

163
Trigonometry, nature of .

1
plane

1

Onity, roots of

175

of the cosine
of the cotangent
of the secant
of the sine
of the tangent

1
72

77
43, 56

22
16
20
21
14
18

.

Variations in the functions
Versed sine

86
171

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