133. Naming trigonometric ratios. - Each of the three constant ratios between the sides of a right triangle that is formed with one of its acute angles equal to a given angle is given a special name. Let a, b, and c, as shown in the figure, be the sides of a right triangle formed with the given acute angle x. Then a C a x b " is called the sine of ∠ x, written sin x, C 6 is called the cosine of <x, written cos x, C a is called the tangent of ∠ x, written tan x. 134. Trigonometric tables. - If a right triangle is accurately constructed, using a protractor, with an acute angle of 40°, and if the sides of this triangle are measured and the ratios of a to c, b to c, and a to b, computed to four decimal places by division, it will be found that Other methods may be employed for computing the trigonometric ratios of angles. Thus, if the angle is 45°, the triangle is isosceles, and hence tan 45° = 1. If the angle is 30°, a = c, and hence sin 30° = 1⁄2 or .5000. 2 The values of the trigonometric ratios of angles have been accurately computed. The following table gives these values for all integral numbers of degrees from 1° to 89°, correct to four decimal places. 135. Finding distances and angles by trigonometric ratios. The trigonometric ratios are used in practical work for the indirect measurement of distances and angles. Thus, to find the distance from A to the B inaccessible point B, an engineer measures a line AC to a convenient point C, making ∠ A a right angle. Then he measures ∠C. If AC = 650 ft. and ∠C = 48°, then tan 48° = 1.1106. AB 650 ft. = A C Hence AB = 1.1106 × 650 ft. = 721.89 ft. 6. Find from the table the angle x if sin x = .3256. If cos x = If tan x = .4663. .1219. 7. In order to find the distance from A to B across a lake, a surveyor measured a line ACat right angles to AB, then measured ∠BCA. If AC = 820 ft. and ∠BCA = 56°, find AB. C 8. The distance from the observer to the foot of a monument is 275 ft. The angle of elevation of the monument at the point of observation is 52°. Find the height. 9. A building known to be 136 ft. high forms an angle of elevation of 23° at a point of observation. How far is the observer from the building? 10. A vertical rod 8 ft. high casts a shadow 3 ft. 6 in. long. Find the angle of elevation of the sun. 11. The distance from the base to the top of a hill, up a uniform incline of 40°, is 300 yd. What is the altitude of the top above the base? 12. A kite string 1000 ft. long makes an angle of 60° with horizontal. How high is the kite, not allowing for sag in the string? It is directly over a spot how far from the holder of the string? 13. A house 32 ft. wide has a gable with rafters 22 ft. long, excluding the part which projects below the eaves. Find the pitch of the roof, or the angle between a rafter and horizontal. 14. The instrument called a telemeter, shown in the drawing, is used by photographers for estimating distances when taking pictures. AB is pointed toward the object E to which the distance is to be found. The plumb line points on the scale to the number of feet that the object E is distant. The instrument is made to be held at a height BD of 5 ft. above the ground. E 15 ft. A FT. B 6-6-15-25-100 C D In making the scale of feet on the telemeter, the 15-ft. mark of the scale must be placed at a point C so that ∠ABC is how many degrees ? Compute in the same way the number of degrees in ∠ABC when Cis at the 6-ft. mark. When Cis at the 25-ft. mark. 15. A mariner lighthouse is 7°. level of the ship. finds that the angle of elevation of a light from a The lighthouse is known to stand 60 ft. above the How far is his ship from the lighthouse? 16. A ship is sailing from New York in a direction 24° north of east. When the log shows that it has gone 450 mi., how far is the ship east of New York? How far north? 24° 17. A sailing ship, tacking against the wind, sails from a port A to B along a course 12° N. of E., from B to C along a course 57° N. of W., then from C to Dalong a course 47° N. of E. The log shows that the ship has gone the following distances: AB = 72 mi., BC = 38 mi., CD = 40 mi. The mariner wishes to know, when at D, how far he is east and how far he is north of A. Compute these distances. SUGGESTION. - Compute AM, BN, and CP. Then add AM and CP and subtract BN from the result to find the distance that D is east of A.. The distance that D is north of A may be found similarly by computing BM, CN, and DP. 19. A 30-ft. flagpole is mounted on the roof of a building. The angles of elevation above horizontal of the top and bottom of the pole are 57° and 50°, respectively. Find the height of the building. SUGGESTION. - Form two equations in two unknown numbers. An equation may be obtained from each right triangle. 30° 45 20. To find the distance from A to B, across a lake, being unable to measure perpendicularly to AB, I measured 120 yd. to C, making ∠CAB = 100°, and observed that ∠ACB = 30°. Find the distance from A to B. SUGGESTION. - First find AD or DB. Then from this 57 C find AB. For finding AD or DB, form two equations in A two unknown numbers. From what two triangles may D they be obtained? 21. How high is a cloud when two observers, so placed as to be in a vertical plane with the cloud, and 880 yd. apart, observe the angles of elevation to be 40° and 55°, respectively? SUGGESTION. - Form two equations in two unknown numbers. 30 ft B |