Ifthe power of sine is odd (n = 2k + 1), save one sine factor and use the identity sin 2 x + cos 2 x = 1 to express the remaining factors in terms of cosine: Let u = cos x then du = – sin x dx. Note: If the powers of both sine and cosine are odd, either of the above methods can be used. Example: Evaluate . Solution: Step 1: Separate one cos ) = cos cos sin sin and sin( ) = sin cos cos sin : Use the Euler identity to show that sin2 + cos2 = 1: Hint: start with ei e i = 1 and use the Euler Identity. B. Wave packets One application of the previous trig identities involves wave interference. We want to use trig identities to rewrite the algebraic sum of two waves, Acos(k 1x ! 1t cossin sin cos cos˚ sin˚ sin˚ cos˚ = cos cos˚ sin sin˚ cos sin˚ sin cos˚ sin cos˚+ cos sin˚ sin sin˚+ cos sin˚ = cos( + ˚) sin( + ˚) sin( + ˚) cos( + ˚) : Since + ˚2R it follows that AB2G. Suppose that A = cos sin sin cos is an element of G, where 2R. Since detA= 1, the inverse of the matrix Ais (see Example 3.26) A 1 = cos Example3 (Finding Exact Values): If sin x = -1/3, and cos y = 2/3, find the exact value of cos (x + y). The number x is an angle in quadrant III, and y is quadrant IV. Don’t use a calculator. Answer . cos (x + y) = (cos x)(cos y) – (sin x)(sin y) • We already know sin x and cos y, however we don’t know cos x and sin y. To find cos x SomeTaylor series, taken about x= 0: ex= X1 n=0 xn n! cos(x) = X1 n=0 ( 1)n x2n (2n)! sin(x) = X1 n=0 ( 1)n x2n+1 (2n+ 1)! 1 1 x = X1 n=0 xn About x= 1: ln(x) = X1 n=1 ( 1)n+1 (x 1)n n. Some integration formulas: R uv0dt= uv R u0vdt; thus R tetdt= tet et+C, R tcos(t)dt= tsin(t) + cos(t) + C, and R tsin(t)dt= tcos(t) + sin(t) + C. Euler’s Thisexample shows how the Fourier series expansion for a square wave is made up of a sum of odd harmonics. Start by forming a time vector running from 0 to 10 in steps of 0.1, and take the sine of all the points. Plot this fundamental frequency. t = 0:.1:10; y = sin (t); plot (t,y); Next add the third harmonic to the fundamental, and plot it. 3L62lMY. wherethe constants mand kare positive. Note that this is a second-order linear (Hint: Try using x= cos(at) or x= sin(at) for some choice of a.) 9. What is the Trigonometrikişlevlerin birim çember üzerinde gösterilmesi. Trigonometrik fonksiyonlar: Sinüs, Kosinüs, Tanjant, Kotanjant, Sekant, Kosekant. ve işlevlerinin kartezyen uzayında grafiksel gösterimi. Trigonometrik fonksiyonlar, matematikte bir açının işlevi olarak geçen fonksiyonlardır. Show using de Moivre's theorem, that sin 5x = 16 sin^ (5) x - 20 sin^ (3) x + 5 sin x. Using the question, first use de Moivre's theorem to say (cos x + i sin x)^5 = cos 5x + isin 5x. Now we can use a binomial expansion on the RHS to say. cos 5x + isin 5x = cos 5 x + 5i cos 4 x sin x +10i 2 cos 3 x sin 2 x + 10i 3 cos 2 x sin 3 x + 5i 4 cos x x= c 1 2cos(t) cos(t) sin(t) + c 2 2sin(t) cos(t) sin(t) : To get the desired initial condition, we take c 1 = 2 and c 2 = 0, so that and Kare measured in some Provingthe derivatives of sin (x) and cos (x) Proving that the derivative of sin (x) is cos (x) and that the derivative of cos (x) is -sin (x). The trigonometric functions and play a significant role in calculus. These are their derivatives:

cos kare x sin kare x