317. The following is a picture of a Hadley's Sextant. AB is the telescope; C' is the fixed mirror; Fis the mirror which can be turned about an axis perpendicular to the plane of ABCF. FN is an arm by which the mirror F is turned. LL' is a graduated arc and N is a vernier for reading its graduations with the aid of the microscope M; K and T are the usual clamp and tangent screw; H is a handle for holding the instrument. At Z are three blackened glasses to interpose between the light and the telescope when looking at a bright object such as the sun; each glass is on a hinge so that it can be brought into the line of sight or turned back at pleasure. At X are blackened glasses which can be placed in the line of the reflected light or turned out of it at pleasure. 318. Thus, the sextant is an instrument for observing the angle subtended at the observer's eye by two distant objects. Its peculiar advantage is that it does not require a steady platform as the Theodolite does. The observer holds it in his hands, and the observation consists in his noting when he has so moved the mirror [EF] that the images of the two distant objects, whose angle he wishes to take, are superposed in the field of the telescope. for This he can do even if he cannot get the images to rest any length of time in the field of view. Accordingly the angle subtended by the edge of the sun and the horizon can with a sextant be observed with considerable accuracy by an observer standing on the deck of a ship in motion. Also, at night, the angular distance between the moon and a star can be observed under like circumstances. 319. Below we give a figure of a Mariner's Compass. It consists of an ordinary magnetic compass with a card attached to the needle, this card is so arranged that when the needle is pointing along the magnetic meridian the pointer on the card is pointing due North. N. The Points of the Compass are figured on the card. They are North. = The angle subtended at the centre of the card by two consecutive points is of 90° = 121°. 253 EXAMPLES FOR EXERCISE. LXXVI a. 1. Define the terms sine, cotangent; and prove that if A be any angle, gin24 + cos24=1. If tan A=2, find sin A and cos A. 2. Trace the changes in the sign and magnitude of cos - sec for values of 0 between 0 and π. 3. Prove geometrically that cos (180° - A)= cos A. Find A if 2 sin A=tan A. 6. Find the greatest side of the triangle of which one side is 2183 feet and the adjacent angles are 78° 14' and 71° 24'. log 2183=3.3390537, L sin 78° 14' = 9.9907766, L sin 30° 22' = 9.7037486, log 42274-4.6260733, 7. Express the other trigonometrical ratios in terms of the cosine. 11. If tan A+ sec 4 = 2, prove that sin A=g, when A is less than 90o. If sin A, prove that tan A+ sec A=3, when A is less than 90o. 12. The length of the greatest side of a triangle is 1035.43 feet, and the three angles are 44°, 66°, and 70°. Solve the triangle, having given L sin 440-9.8417713, L sin 700-9.9729858, log 765432=5.8839067, L sin 66° 9.9607302, log 1035 43=3.0151212, log 10066=4.0028656. 13. Express the other trigonometrical ratios in terms of the cotangent. 15. Write down the tangents of all the angles which are multiples of 30° and less than 360o. 16. If tan A+ sec A=3, prove that sin A= when A is less than 90o. If sin 4%, prove that tan A+ sec A =2, when A is less than 90o. 17. Find the sines of the three angles of the triangle whose sides are 193, 194, and 195 feet. 18. Investigate the following formulæ : 21. Find a formula to include all angles that have the same cotangent as the angle a. Solve the equation tan 0=cot 0. 22. Prove the formula to express the cosine of the sum of two angles in terms of the sines and cosines of those angles. Express cos 5a in terms of cos a. |