Trigonometry Functions (5B) Young Won Lim 7/24/14

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Transcript:

Trigonometry Functions (5B 7/4/14

Copyright (c 011-014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions to youngwlim@hotmail.com. This document was produced by using OpenOffice and Octave. 7/4/14

Trigonometric Functions http://en.wikipedia.org/wiki/derivative Trigonometric Function (5B 3 7/4/14

Radian http://en.wikipedia.org/wiki/derivative Trigonometric Function (5B 4 7/4/14

Pythagorean identity http://en.wikipedia.org/wiki/derivative Trigonometric Function (5B 5 7/4/14

Inverse Relations http://en.wikipedia.org/wiki/derivative Trigonometric Function (5B 6 7/4/14

Symmetry http://en.wikipedia.org/wiki/derivative Trigonometric Function (5B 7 7/4/14

Shifts and periodicity http://en.wikipedia.org/wiki/derivative Trigonometric Function (5B 8 7/4/14

Angle sum and difference identities http://en.wikipedia.org/wiki/derivative Trigonometric Function (5B 9 7/4/14

Double Angle Formula http://en.wikipedia.org/wiki/derivative Trigonometric Function (5B 10 7/4/14

Triple-angle formulae http://en.wikipedia.org/wiki/derivative Trigonometric Function (5B 11 7/4/14

Half-angle formulae http://en.wikipedia.org/wiki/derivative Trigonometric Function (5B 1 7/4/14

Power-reduction formula http://en.wikipedia.org/wiki/derivative Trigonometric Function (5B 13 7/4/14

Product-to-sum http://en.wikipedia.org/wiki/derivative Trigonometric Function (5B 14 7/4/14

Sum-to-product http://en.wikipedia.org/wiki/derivative Trigonometric Function (5B 15 7/4/14

Euler's Formula e i = cos( + i sin( i( A +B e = cos(a +B + i sin (A +B e i A e i B = (cos(a + i sin( A(cos(B + i sin(b = [cos(acos(b sin( Asin(B] + i[cos( Asin(B + sin( Acos(B] sin (A +B = sin( Acos(B + cos( Asin (B cos( A+B = cos( Acos(B sin( Asin(B Trigonometric Function (5B 16 7/4/14

Sin( angle sum and difference sin (A +B sin (A sin (B sin (Acos(B cos( Asin (B cos( A A cos(b B sin (A B sin (A sin (B sin (Acos(B cos( Asin (B cos( A A cos(b B Trigonometric Function (5B 17 7/4/14

Cos( angle sum and difference cos( A+B sin (A sin (B cos( Acos(B sin( Asin (B cos( A A cos(b B cos( A B sin(a sin (B cos( Acos(B + sin( Asin(B cos( A A cos(b B Trigonometric Function (5B 18 7/4/14

Product to Sum : sin cos sin (A +B +sin (A B sin( A sin (B sin(a sin(b cos( A cos(b cos( A cos(b A B A B sin( A+B + sin( A B sin( Acos(B Trigonometric Function (5B 19 7/4/14

Product to Sum : cos sin sin( A+B sin( A B cos( Asin(B sin (A +B sin( A B sin (A sin (B sin(a sin(b cos( A cos(b cos( A cos(b A B A B Trigonometric Function (5B 0 7/4/14

Product to Sum : cos cos cos( A+B +cos( A B sin (A sin (B sin(a sin(b cos( A cos(b cos( A cos(b A B A B cos( A +B + cos( A B cos( Acos(B Trigonometric Function (5B 1 7/4/14

Product to Sum : sin sin cos( A +B cos( A B sin( Asin(B cos( A+B + cos( A B +sin( Asin(B cos( A+B cos( A B sin (A sin (B sin(a sin(b cos( A cos(b cos( A cos(b A B A B Trigonometric Function (5B 7/4/14

Product to Sum sin (A +B +sin (A B sin( A sin (B sin(a sin(b cos( A cos(b cos( A cos(b A B A B sin (A +B sin( A B sin (A sin (B sin(a sin(b cos( A cos(b cos( A cos(b A B A B Trigonometric Function (5B 3 7/4/14

Product to Sum cos( A+B +cos( A B sin (A sin (B sin(a sin(b cos( A cos(b cos( A cos(b A B A B cos( A+B cos( A B sin (A sin (B sin(a sin(b cos( A cos(b cos( A cos(b A B A B Trigonometric Function (5B 4 7/4/14

Sum and Difference A A+B = X B A B = Y X +Y = A+B+ A B = A X Y = A+B A+B = B Trigonometric Function (5B 5 7/4/14

Product to Sum sin (X +sin (Y sin ( X +Y sin ( X Y sin ( X +Y sin ( X Y cos( X +Y cos( X Y cos( X +Y cos( X Y X +Y X Y X +Y X Y sin (X sin(y sin ( X +Y sin ( X Y sin ( X +Y sin ( X Y cos( X +Y cos( X Y cos( X +Y cos( X Y X +Y X Y X +Y X Y Trigonometric Function (5B 6 7/4/14

Product to Sum cos(x +cos(y sin ( X +Y sin ( X Y sin ( X +Y sin ( X Y cos( X +Y cos( X Y cos( X +Y cos( X Y X +Y X Y X +Y X Y cos(x cos(y sin ( X +Y sin ( X Y sin ( X +Y sin ( X Y cos( X +Y cos( X Y cos( X +Y cos( X Y X +Y X Y X +Y X Y Trigonometric Function (5B 7 7/4/14

Product-to-Sum & Sum-to-Product SUM sin( A+B + sin( A B sin( Acos(B sin( A+B sin( A B cos( Asin(B cos( A +B + cos( A B cos( Acos(B cos( A+B + cos( A B sin( Asin(B SUM PRODUCT PRODUCT sin(x + sin(y sin( X +Y cos( X Y sin(x sin(y cos( x+y X Y sin( cos(x + cos(y cos( X +Y cos( X Y cos( X + cos(y sin( X +Y sin( X Y Trigonometric Function (5B 8 7/4/14

Derivatives http://en.wikipedia.org/wiki/derivative Trigonometric Function (5B 9 7/4/14

Unit Circle Geometry 0 < < π/ = l π r π = l r = l (rad r = 1 sin ( tan( sin( < < tan( 1 sin ( tan( = sin ( cos ( 1 tan( cos( 1 Trigonometric Function (5B 30 7/4/14

Inequalities 0 < < π/ = l π r π = l r = l (rad r = 1 sin ( tan( sin( < < tan( sin( < 1 < tan( Trigonometric Function (5B 31 7/4/14

Sinc(x 0 < sin ( < 1 0 < sin ( < 1 1 < sin( < 1 if 0 1 < sin( < 0 < 1 1 < sin ( < 0 1 0.8 0.6 0.4 0. 0-0. -0.4 0 0 40 60 80 100 Trigonometric Function (5B 3 7/4/14

Sin(x / x 0 < 1 = l π r π = l r = l (rad r = 1 sin ( tan( sin( < < tan( lim 0 sin( = 1 sin( < 1 < tan( cos( < sin ( < 1 cos( < sin ( 1 < tan( Trigonometric Function (5B 33 7/4/14

(1 cos(x / x 0 < < π/ 0 < 1 r = 1 r = 1 cos( 1 cos( 1 cos( 1+cos( 1+cos( = 1 cos ( (1+cos( 1 cos( sin( 1 cos( = sin( lim 0 1 sin ( (1+cos( 1 cos( = 0 Trigonometric Function (5B 34 7/4/14

sin(x / x, cos(x / x, (1 cos(x / x sin ( cos( 1 cos( Trigonometric Function (5B 35 7/4/14

The Derivative of the Sine Function d d x f (x = lim h 0 f (x+h f (x h d d x sin(x = lim h 0 sin(x+h sin(x h = lim h 0 = lim h 0 = sin(xlim h 0 = cos(x sin (xcos(h + cos( xsin(h sin( x h sin (x(cos(h 1 + cos( xsin(h h (cos(h 1 h + cos(xlim h 0 sin (h h Trigonometric Function (5B 36 7/4/14

The Derivative of the Cosine Function d d x f (x = lim h 0 f (x+h f (x h d d x cos(x = lim h 0 cos(x +h cos(x h = lim h 0 = lim h 0 = cos(xlim h 0 = sin(x cos(xcos(h sin (xsin(h cos( x h cos(x(cos(h 1 sin (xsin (h h (cos(h 1 h sin( xlim h 0 sin(h h Trigonometric Function (5B 37 7/4/14

The Derivative of the Tangent Function d d x f (x = lim h 0 f (x+h f (x h d d x tan(x = d ( sin(x d x cos(x = [sin(x]' cos(x sin(x[cos(x]' cos (x = cos (x+sin (x cos (x = 1 cos (x = sec (x Trigonometric Function (5B 38 7/4/14

Derivative of sin(x f (x = sin(x +1 0-1 0 +1 0-1 0 +1 0-1 0 slope leads d d x f (x = cos(x Trigonometric Function (5B 39 7/4/14

Derivative of cos(x f x = cos x 0-1 0 +1 0-1 0 +1 0-1 0 +1 slope leads d d x f x = sin x Trigonometric Function (5B 40 7/4/14

arcsin(x Trigonometric Function (5B 41 7/4/14

arccos(x Trigonometric Function (5B 4 7/4/14

arctan(x Trigonometric Function (5B 43 7/4/14

Integration Trigonometric Function (5B 44 7/4/14

Inverse Relations http://en.wikipedia.org/wiki/derivative Trigonometric Function (5B 45 7/4/14

References [1] http://en.wikipedia.org/ [] M.L. Boas, Mathematical Methods in the Physical Sciences [3] E. Kreyszig, Advanced Engineering Mathematics [4] D. G. Zill, W. S. Wright, Advanced Engineering Mathematics 7/4/14

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