Trigonometry Tidbits
Discover Euler's formulas, equations, and insights on complex numbers in this in-depth workshop focused on enhancing trigonometry teaching. Uncover the beauty of mathematical identities and expand your understanding of complex numbers through engaging content and visual aids.
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Presentation Transcript
Trigonometry Tidbits Teacher Quality Workshop 2/16/2017
Eulers Formula 2 4 6 3 5 7 = + + = + + cos 1 ... sin ... 2! 4! 6! 3! 5! 7! = = 0 1 i 2 3 4 e = + + + + + 1 ... 1! 2! 3! 4! 1 i i ( ) ( ) 1! i i + ( ) 3! 3 ( ) 4! 4 = = 2 2 3 4 1 i i i i i = + + + + + i 1 ... e 2! 2 = 3 2 i i i i 2 3 4 5 5 6 6 i i i i = + + + + + 1 ... = = = = 4 3 1 i i i 1! 2! 3! 4! 5! 6! 5 4 i i i i i 2 3 4 5 6 i i = + + + 1 . .. 1! 2! 3 ! 4! 5! 6! Cycle continues
Eulers Neat Formula i 2 3 4 5 6 i i = + + + i 1 ... e 1! 2! 3! 4 ! 5 ! 6 ! 2 4 6 3 5 = + + + + + ie 1 ... ... i 2! 4! 6! 1! 3! 5! 2 4 6 3 5 7 = + + = + + cos 1 ... sin ... 2! = 4! 6! 3! 5! 7! ( ) + ie co s s in i
Equality of Complex Numbers Complex numbers w and z are equal if and only if Re(w) = Re(z) and Im(w) = Im(z). ----------------------------------------------------------- a + i b = c + i d a = c and b = d
Eulers Formula Unlocks Trig Identities Expand (ei )2two different ways 1. First use law of exponents, then apply Euler s Formula 2. First use Euler s Formula, then multiply out
Expand (ei)2 1. First use law of exponents, then apply Euler s Formula
Expand (ei)2 2. First use Euler s Formula, then multiply out
Set the two expansions of equal (ei)2equal Replace 2 with a:
Eulers Formula Unlocks Trig Identities Expand (eiA) (eiB) two different ways 1. First use law of exponents, then apply Euler s Formula 2. First use Euler s Formula, then multiply out
Expand (eiA) (eiB) 1. First use law of exponents, then apply Euler s Formula
Expand (eiA) (eiB) 2. First use Euler s Formula, then multiply out
Set the two expansions of equal (eiA) (eiB) equal Replace B with B, use even/odd properties:
Sine strikes the right cord. Sanskrit word jiva , meaning bowstring Arabic translation jayb also meaning bosom of a garment Latin translation sinus English translation sine https://www.macalester.edu/ ~bressoud/talks/2011/Histor y0fTrig_NHCC.pdf http://mathforum.org/library /drmath/view/52578.html
Why is the tangent function called tangent? See the Geogebra app called tangent_function.ggb
Why is the secant function called secant? See the Geogebra app called secant_function.ggb
The whole shebang in one picture Cosine, cotangent, and cosecant are the complementary functions.
The Complementary Functions + = 90 = cos sin = cot tan = csc sec
Set the two expansions of equal (ei)2 equal + = + 2sin cos 2 2 cos2 sin2 cos sin i i = 2 2 cos2 cos sin = 2sin cos sin2 Replace 2 with a: a a = 2 2 cos cos sin a 2 2 a a = sin 2sin cos a 2 2
Set the two expansions of equal (eiA) (eiB) equal ( ) ( ) ( sin cos cos cos i A B A B + ( ) cos cos cos A B A + = ( ) sin sin A B A + = + + + = + cos sin A B i A B ) cos sin A sin s in B A B sin sin B A B + cos cos sin A B B Replace B with B, use even/odd properties: ( ) cos cos cos A B = ( ) sin sin A B = + sin sin A B A B cos cos sin A A B B
