Trigonometric Identities - Miami
2 cos x+y 2 cosx+ cosy= 2cos x+y 2 cos x y 2 cosx cosy= 2sin x+y 2 sin x y 2 The Law of Sines sinA a = sinB b = sinC c Suppose you are given two sides, a;band the
Trigonometric Limits
sin(x) = 0, lim x→0− (1 − cos(x)) = 0 The left and the right limits are equal, thus lim x→0 sin(x) = 0, lim x→0 (1 − cos(x)) = 0 or, lim x→0 sin(x) = 0, lim x→0 cos(x) = 1 – Typeset by FoilTEX – 8
Euler’s Formula and Trigonometry
satisfying x2 + y2 = 1, we have cos2 + sin2 = 1 Other trignometric identities re ect a much less obvious property of the cosine and sine functions, their behavior under addition of angles This is given by the following two formulas, which are not at all obvious cos( 1 + 2) =cos 1 cos 2 sin 1 sin 2 sin( 1 + 2) =sin 1 cos 2 + cos 1 sin 2 (1)
Trigonometric Identities
cos2 = cos2 sin2 = 2cos2 1 = 1 2sin2 (10) sin2 = 2sin cos (11) tan2 = 2tan 1 tan2 (12) Note that you can get (5) from (4) by replacing B with B, and using the fact
Fourier Series - Bard
2 cos(A B) + 1 2 cos(A+ B) sinAcosB = 1 2 sin(A B) + 1 2 sin(A+ B) sinAsinB = 1 2 cos(A B) 1 2 cos(A+ B): These identities allow us to transform any product of trigonometric functions into a sum By applying them repeatedly, we can remove all of the multiplications from a trigonometric polynomial, resulting in a Fourier sum
Integral formulas for Fourier coefficients
2 4 ˇ X1 n=0 cos(0) (2n + 1)2 = ˇ 2 4 ˇ X1 n=0 1 (2n + 1)2 Solving for the series gives the result Remark: In Calculus II you learned that this series converges, but were unable to obtain its exact value Daileda Fourier Coe cients
CHAPTER 4 FOURIER SERIES AND INTEGRALS
Multiply both sides of (2) by 2/π: Sine coefficients S(−x)=−S(x) b k = 2 π π 0 S(x)sinkxdx= 1 π π −π S(x)sinkxdx (6) Notice that S(x)sinkx is even (equal integrals from −π to 0 and from 0 to π) I will go immediately to the most important example of a Fourier sine series S(x) is an odd square wavewith SW(x)=1for0
Commonly Used Taylor Series
1 x = 1 + x + x2 + x3 + x4 + ::: note this is the geometric series just think of x as r = X1 n=0 xn x 2( 1;1) ex = 1 + x + x2 2 + x3 3 + x4 4 + ::: so: e = 1 + 1 + 1 2 + 3 + 1 4 + ::: e(17x) = P 1 n=0 (17 x)n = X1 n=0 17n n n = X1 n=0 xn n x 2R cosx = 1 x2 2 + x4 4 x6 6 + x8 8::: note y = cosx is an even function (i e , cos( x
1 Lecture 24: Linearization - University of Kentucky
Example Suppose that a curve is given by the equation x2 + y3 = 2x2y Verify that the point (x;y) = (1;1) lies on the curve Assume that the curve is given by a function y= y(x) for xnear 1 and approximate y(1:2) Solution To verify that (x;y) = (1;1) lies on the curve, we need to know that 13 + 12 = 2 12 1 which is true
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