10. A flagstaff which leans to the east is found to cast shadows of 198 ft. and 202 ft., when the sun is due east and west respectively, and his altitude is 7°. Find the length of the flagstaff and its inclination to the vertical.

11. What angle will a flagstaff 24 ft. high, on the top of a tower 200 ft. high, subtend to an observer on the same level with the foot of the base, and 100 yds. distant from it?

12. Looking out of a window with his eye at the height of 15 ft. above the roadway, an observer finds that the angle of elevation of the top of a telegraph post is 17° 18' 35', and that the angle of depression of the foot of the post is 8°32′ 15′′. Calculate the height of the telegraph post and its distance from the observer.

13. A man in a balloon, when it is one mile high, finds the angle of depression of an object on the level ground to be 35° 20', then after ascending vertically and uniformly for 20 min., he finds the angle of depression of the same object to be 55° 40'. Find the rate of ascent of the balloon in miles

per hour.

14. A man observes the elevation of a mountain top to be 15°, and after walking 3 mi. directly toward it on level ground, the elevation is 18°. Find his distance from the mountain.

15. From a boat the angle of elevation of the highest and lowest points of a flagstaff, 30 ft. high, on the edge of a cliff are observed to be 46° 12' and 44° 13'. Determine the height of the cliff and its distance.

16. The angles of elevation of the top of a tower, observed at two points in the horizontal plane through the base of the tower, are tan-1 and tan-1; the points of observation are 240 ft. apart, and lie in a direct line from the base. Find the height of the tower.

17. A person standing due south of a lighthouse observes that his shadow cast by the light at the top is 24 ft. long; on walking 100 yd. due east he finds his shadow to be 30 ft. Supposing him to be 6 ft. high, find the height of the light from the ground.

18. An observer is 384 yd. due south of a point from which a balloon ascended; he measures a horizontal base due east, and at the other extremity finds the angle of elevation to be 60° 15'. Find the height of the balloon.

19. A surveyor starts from A and runs 766 yd. due east to B, thence 622 yd. N. 20° 30' E. to C, thence 850 yd. N. 41° 45′ W. to D, thence S. 42° 35′ W. to E. Find the distance and bearing of A from E, and determine the area of the field ABCDE.

20. A surveyor runs 253 yd. N.E. by E., thence N. by E. 212 yd., thence W.N. W. 156 yd., thence S. W. by S. 210 yd., thence to the startingpoint. Find the bearing and distance of the starting-point from the last station, and determine the area of the field which the surveyor has gone around.

CHAPTER V.

1. Define and illustrate angle, negative angle, complement of an angle, supplement of an angle, quadrant, angle in the third quadrant.

2. Define and illustrate the six trigonometric ratios. Find the greatest and least values that each of them can have. Arrange in tabular form the algebraic signs of the trigonometric ratios of an angle in each quadrant.

3. Explain how the trigonometric ratios of an angle of any magnitude, positive or negative, can be found, (a) by means of tables which give these ratios for angles up to 90° only, (b) by means of tables which give these ratios for angles up to 45° only.

4. Prove that if two angles have the same sine, and also any of the other five trigonometric ratios (with one exception) the same, they will differ by a multiple of 360°.

5. State and prove the chief relations which exist between the trigonometric ratios of any angle A.

A,

6. Express the trigonometric ratios of 90° A, 90° A, 180° 180° + A, 270° – A, 270° + A, 360° — A, — A, in terms of the trigonometric ratios of A.

7. Name three pairs of trigonometric ratios such that the product of each pair shall equal 1; one pair, the sum of whose squares shall equal 1; two pairs, the difference of whose squares shall equal 1.

8. Compare the trigonometric ratios of any angle (a) with those of its complement, (b) with those of its supplement.

10. (a) Express the following trigonometric ratios in terms of trigonometric ratios of positive angles not greater than 45° : sin 237°, cos (— 410°), tan 2000°, cot (— 137°), sec 445°, cosec (— 650°), sin 185°, tan 267°, sec 345°, cos 87°, cot (- 19°); (b) by means of the tables give the numerical values of these ratios.

11. Find, without the use of trigonometric tables, the numerical values of cos 1410°, tan (- 1260°), cosec (-1710°), tan 225°, cot 1035°, cosec 210°, cos 1500°, sin 1665°, tan (- 1665°), all the trigonometric ratios of and 930°.

1125°

12. Construct the angles: (a) whose secant is 3, (b) whose tangent is √2+1, (c) whose cotangent is . Find the other ratios of these angles.

13. (a) Find sin A, cot A, when cos A the other ratios of A and x when cot A other ratios of A when cos A =

-

(b) Find

, and

, and A < 180°. and cos x = — g. (c) Find the A lies between 540° and 630°.

5

(d) Find the trigonometric ratios of 180° + 0 and 270° – 0, given tan 0 = (e) Given sec x = — , and x in the third quadrant; find the value of sin x+tan x

cos x + cot

14. Do Ex. 9, Art. 18, A being any angle. Explain the ambiguities in the algebraic signs. If A is an angle in the third quadrant, express cos A, tan A, cot A, sec A, cosec A in terms of sin A.

15. (a) If sec A = n tan A, find the other ratios of A. (b) If 2 sec 0 = tana+cot α, find tan 0 and cosec 0. (c) Solve x3 cot 108° = 128° sin 72° cos 18°. 16. Prove the identities: sin3 0 + cos3 0 = (sin 0 + cos 0) (1 − sin cos 0); cost A sin1 A = 1 − 2 sin2 A; sin x (cot x + 2) (2 cot x + 1) = 2 cosec x + 5 cos x; sec2 B - cos2 B = cos2 B tan2 B+ sin2 B sec2 B; cos® A + sino A = 1 - 3 cos2 A sin2 A; cos6 x + 2 cost x sin2 x + cos2 x sin1 x + sin2 x = 1.

17. (a) Find the value of x not greater than two right angles which will satisfy the equation 4√3 cot x=7 cosec x-4 sin x. (b) Likewise, in the case of the equation sin x+cosx cotx=2. (c) Likewise, in tan1 x-4 tan2x+3=0. (d) If 1 + sin2 0 = 3 sin 0 cos 0, find tan 0. (e) Find the least positive value of A that satisfies the equation 2√3 cos2 A = sin A. (ƒ) Find all the angles between 0° and 500° which satisfy the equation 4 sin2 0 = 3. (g) If 2 cos A+ sec A = 3, what is the value of A? (h) Find A when tan2 A + cosec2 A = 3.

CHAPTER VI.

1. (a) Write the values of cos (A+B), cos (A – B), sin(A + B), sin (A – B), tan (A + B), tan (A – B) in terms of the trigonometric ratios of A and B. (b) Deduce these values. (c) Express them in words.

2. (a) Express in terms of the trigonometric ratios of A each of the following: sin 2 A, cos 2 A (three different forms), tan 2 A, cot 2 A. (b) Derive these expressions.

COS

A+ B A-B
2

2

3. (a) Show that sin A+ sin B = 2 sin (b) Show that cos 2 A + cos 2 B = 2 cos (A + B) cos (A – B). (c) State and derive an equivalent expression for the difference of two sines; (d) for the difference of two cosines.

4. Show how to find cos A when cos A is known. Explain the ambiguity in the result. Determine the sign of the result when A is an angle in the third quadrant. Find the cosine of 112° 30'.

5. Prove 2 cos

and 360°.

==

2

√1 + sin A - √1-sin A if A

[From cos 225°.]

is between 270°

6. Derive an expression for each of the following: sin 3 A in terms of sin A, cos 3 A in terms of cos A, tan 3 ▲ in terms of tan A. [SUGGESTION:

(b) Prove

9. (a) Find sin 45°, and thence deduce the ratios of 22° 30'. that tan 67° 30′ =1+ √2. (c) Deduce the ratios of 67° 30', (i.) from ratios of 45°, 22° 30', (ii.) from ratios of 135°.

=

10. (a) Given sin 30° and cos 45° = √2; find sin 15°, cos 75°. (b) Given sin 30°; find the numerical values of the other ratios of 30°; thence derive the ratios of 15°, thence derive the ratios of 75°, 105°, 165°, 195°. (c) Prove the following: tan 15° + tan 75° = 4, cos 15° · cos 75° = .25, sin 105°+cos 105°-cos 45°, tan 15° (tan 60°-tan 30°) = tan 60°+tan 30°−2.

11. (a) Express sin 8 A + sin 2 A as a product. (b) Express as a sum or difference: (i.) 2 cos A cos B, (ii.) 2 sin 50° cos 20°. (c) Prove without using tables that (i.) sin 70° sin 10° = cos 40°, (ii.) cos 20° + cos 100° +

cos 140° = 0. Verify by the tables.

=

12. Show that: (1) cot A cot B cos(A+B)=cos A cos B(cot A cot B-1); (2) cos (A + B) cos A + sin (A + B) sin A = cos B; (3) cos A sin A = √2 cos(A+45°); (4) 2 cos x — 2 sin3 x = cos 2 x(1 + cos2 2x); (5) cos2 A+ sin2 A cos 2 B = cos2 B+ sin2 B cos 2 A; (6) cos2 A-cos A cos(60° + A)+ sin2 (30° - A) = .75; (7) tan 10 sin 0:1 + cos 0; (8) cos (135° + A) + sin (135° — A)=0; (9) cosec 20+ cot 2 0 = cot 0. 13. Prove that: 2 sin A sin 2 A 2 sin A+ sin 2 A

(1)

sin x + sin y

==

cot (x − y);

(2) tan

A

COS X

-

2

8 cot A

(3) tan (60° + A) — tan (60° − A) :

=

cot2 A 3 cos 2 B-cos 2 A sin 2 A-sin 2 B sin 2 B+ sin 2 A cos 2 A+ cos 2 B tan 2 A sin 2 A − cot2 A + tan2 A cot2 A tan2 A

==

14. (a) Find values of not greater than 180°, which satisfy cot 0=tan

(b) Give all the positive angles less than 360°, which satisfy the equation sin 2 A = √3 cos 2 A.

15. Show that the value of sin (n+1)B sin(n−1)B+cos(n+1)B cos(n−1)B is independent of n.

16. The cosines of two angles of a triangle ABC are and 1, respectively; find all the trigonometric ratios of the third angle without using tables. Verify the results by means of the tables.

4

17. Two towers whose heights respectively are 180 and 80 ft., stand on a horizontal plane; from the foot of each tower the angle of elevation of the other is taken, and one angle is found to be double the other; prove that the horizontal distance between the towers is 240 ft., and show that the sine of the greater angle of elevation is .6.

==

CHAPTER VII.

1. In a triangle ABC, show that (1) sin(A + B) = sin C, (2) cos(A + B) cos C, (3) sin

A+ B
2

C
= COS , (4) cos

A+ B
2

= sin ene

C

2. (a) State and prove the Law of Sines for the plane triangle. (b) State and prove the Law of Cosines for the plane triangle. (c) If the sines of the angles of a triangle are in the ratios of 13: 14 : 15, prove that the cosines are in the ratios of 39:33:25.

3. (a) Prove that in ABC, b + c : b - c = tan (B+ C) : tan † (B − C') =cot A: tan (B-C). (b) Write and derive the expressions for the cosine of an angle of a triangle, and the cosine and the sine of half that angle, in terms of the sides of the triangle. (c) In the triangle ABC derive the formulas expressing tan 1⁄2 A, tan 1⁄2 B, tan 1⁄2 C, in terms of a, b, c. (d) Prove that in any triangle ABC, sin A √s(s − a) (s — b) (s — c).

2

bc

4. (a) Show how to solve a triangle when the three sides are given, (i.) without logarithms, (ii.) with logarithms. Derive all the formulas necessary. (b) Do the same when two sides and their included angle are given. (c) Do the same when two angles and a side are given.

5. (a) Explain carefully, and illustrate by figures, the case in which the solution of a triangle is ambiguous. (b) Write formulas for a complete solution and check, of a triangle, when two sides and an angle opposite to one of them are given. How many solutions are there? Discuss fully all cases that may arise. (c) Given the angle A, and the sides a and b of a triangle ABC, determine whether there will be one solution, two solutions, or no solution, in each of the following cases: (i.) A < 90°, a>b, (ii.) A<90°, a = b, (iii.) A< 90°, a <b, (iv.) A>90°, a >b, (v.) A> 90°, a = b.

6. Show by the trigonometric formulas that the angles of a triangle can be found when the ratios of the three sides are given. Give the geometrical explanation.

7. Show by the trigonometric formulas that the other two angles of a triangle can be found when the third angle, and the ratio of the sides containing it, are known. Give the geometrical explanation.