Plane Trigonometry and Tables

Exercise 84. Review Problems

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 58° and 63° respectively. Find the height of the aeroplane and its distance from each observer. 3. Prove that (Vcsc x + cotx cot x)2 = 2 (csc x − 1). 4. Given sin x = 2 m/(m2 + 1) and sin y = 2 n/(n2 + 1), find the value of tan (x + y).

5. Find the least value of cos2x + sec2x.

6. Prove that 1- sin x/sin2y = cos2x (1-
cos2x (1 — tan2x/tan2y).

7. Prove this formula, due to Euler: tan-tan-1} = π. 8. Prove that cotx cot x csc x.

9. Prove that (sin x+i cos x)" = cos n († π − x) + i sin n (π— x).

10. Show that log iTi and that log (-i) = Ti.

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 4r cot A cot B cot C, where r is 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 — § a2]+.

Prove the following formulas:

15. 2 cos2x = cos 2 x + 1.

16. 2 sin2x