SECTION 6.2 Trigonometric Functions: Unit Circle Approach 407 Notice that the triangles OA P* * and OAP are similar; as a result, the ratios of corresponding sides are equal. = = = = = = y y r x x r y x y x y r y x r x x y x y * 1 * 1 * * 1 * 1 * * * These results lead us to formulate the following theorem: THEOREM For an angle θ in standard position, let P x y , ( ) = be the point on the terminal side of θ that is also on the circle x y r . 2 2 2 + = Then θ θ θ θ θ θ = = = ≠ = ≠ = ≠ = ≠ y r x r y x x r y y r x x x y y sin cos tan 0 csc 0 sec 0 cot 0 Finding the Exact Values of the Six Trigonometric Functions Find the exact values of each of the six trigonometric functions of an angle θ if 4, 3 ( ) − is a point on its terminal side in standard position. Solution EXAMPLE 12 See Figure 36. The point 4, 3 ( ) − is on a circle with center at the origin that has a radius of r 4 3 16 9 25 5. 2 2 ( ) = + − = + = = For the point x y , 4, 3 , ( ) ( ) = − we have x 4 = and y 3. = − Since r 5, = we find θ θ θ θ θ θ = = − = = = = − = = − = = = = − y r x r y x r y r x x y sin 3 5 cos 4 5 tan 3 4 csc 5 3 sec 5 4 cot 4 3 Now Work PROBLEM 77 Figure 37 a B D C A O Historical Feature The name sine for the sine function arose from a medieval confusion. Its origin is from the Sanskrit word jı– ba (meaning “chord”), first used in India by Araybhata the Elder ( AD 510). He really meant half-chord, but abbreviated it. This was brought into Arabic as jı– ba , which was meaningless. Because the proper Arabic word jaib would be written the same way (short vowels are not written out in Arabic), jı– ba was pronounced as jaib , which meant “bosom” or “hollow,” and jaib remains as the Arabic word for sine to this day. Scholars translating the Arabic works into Latin found that the word sinus also meant “bosom” or “hollow,” and from sinus we get sine . The name tangent , due to Thomas Finck (1583), can be understood by looking at Figure 37. The line segment DC is tangent to the circle at C. If d O B d O C , , 1, ( ) ( ) = = then the length of the line segment DC is α ( ) ( ) ( ) ( ) = = = d D C d D C d D C d O C , , 1 , , tan The old name for the tangent is umbra versa (meaning turned shadow), referring to the use of the tangent in solving height problems with shadows. The names of the remaining functions came about as follows. If α and β are complementary angles, then cos sin . α β = Because β is the complement of ,α it was natural to write the cosine of α as sin co .α Probably for reasons involving ease of pronunciation, the co migrated to the front, and then cosine received a three-letter abbreviation to match sin, sec, and tan. The two other cofunctions were similarly treated, except that the long forms cotan and cosec survive to this day in some countries. Figure 36 6 6 (4, 23) 26 26 y x r 5 5 x2 1 y2 5 25 u
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