# New PDF release: Die Geometrie der Beruehrungstransformationen

By Lie Sophus, Scheffers G.

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Extra info for Die Geometrie der Beruehrungstransformationen

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35 PLANE TRIGONOMETRY 36 In general we call a line segment positive if it generated by a point moving from left to right, as AB Figure 15. We indicate such a line segment by AB. distances. is in Figure 16. the line segment is generated by a point moving from right to left, we call the line segment negative, and indicate If it by BA. ) Thus, to B, in Figure 15, BA = -AB. But it is desirable often to distinguish between positive and negative distances when the direction is neither from left to right nor from right to left.

Functions of ( 0) . What relations hold between the functions of a negative angle and the functions of the corresponding positive angle? Consider Figure 28, FUNCTIONS OF ALL ANGLES r x y - r' = x = -y' *---V I/ sin (-0) 0) ( = -x = x r r cot \$ = +cos = esc = sec ( 0) = -cot (-0) 0) ( sec - _ sin r t/ v = - = -^v = -- = -tan tan (~0) csc H r r' cos 49 Problems Example Find: sin - 1 300. sin - = -sin 300 - -sin (360 - 60) = -(-sin 60) 300 sin 60 V3 as "^' Example 2 Find all - 2 sin 2 and 360 angles between 1 = 0.

Sin x + (g) sin 2x 2 (h) 2 sin (fc) 4. -\ = 2) 0. = 0. sin x. + 5 cos x 6 = 0. + 1 = 0. unknown V THE OBLIQUE TRIANGLE CHAPTER 24. Law of sines. In Chapter III we considered the solution of the right triangle. In this chapter we shall be concerned with the solution of the oblique triangle, for which we use two important laws giving relations between the sides and the angles of such a triangle. We proceed to the derivation of the first of these laws, called the law of sines: Law of Sines. The sides of a triangle are proportional to the sines of the angles opposite.