Young Won Lim7/24/14

Trigonometry Functions (5B)

Young Won Lim7/24/14

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Trigonometric Function (5B) 3 Young Won Lim7/24/14

Trigonometric Functions

http://en.wikipedia.org/wiki/Derivative

Trigonometric Function (5B) 4 Young Won Lim7/24/14

Radian

http://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Derivative

Trigonometric Function (5B) 5 Young Won Lim7/24/14

Pythagorean identity

http://en.wikipedia.org/wiki/Derivative

Trigonometric Function (5B) 6 Young Won Lim7/24/14

Inverse Relations

http://en.wikipedia.org/wiki/Derivative

Trigonometric Function (5B) 7 Young Won Lim7/24/14

Symmetry

http://en.wikipedia.org/wiki/Derivative

Trigonometric Function (5B) 8 Young Won Lim7/24/14

Shifts and periodicity

http://en.wikipedia.org/wiki/Derivative

Trigonometric Function (5B) 9 Young Won Lim7/24/14

Angle sum and difference identities

http://en.wikipedia.org/wiki/Derivative

Trigonometric Function (5B) 10 Young Won Lim7/24/14

Double Angle Formula

http://en.wikipedia.org/wiki/Derivative

Trigonometric Function (5B) 11 Young Won Lim7/24/14

Triple-angle formulae

http://en.wikipedia.org/wiki/Derivative

Trigonometric Function (5B) 12 Young Won Lim7/24/14

Half-angle formulae

http://en.wikipedia.org/wiki/Derivative

Trigonometric Function (5B) 13 Young Won Lim7/24/14

Power-reduction formula

http://en.wikipedia.org/wiki/Derivative

Trigonometric Function (5B) 14 Young Won Lim7/24/14

Product-to-sum

http://en.wikipedia.org/wiki/Derivative

Trigonometric Function (5B) 15 Young Won Lim7/24/14

Sum-to-product

http://en.wikipedia.org/wiki/Derivative

Trigonometric Function (5B) 16 Young Won Lim7/24/14

Euler's Formula

e iθ= cos(θ) + i sin(θ)

e i(A+B) = cos(A+B) + i sin (A+B)

e i A ei B = (cos(A) + i sin(A))(cos(B) + i sin (B))

= [cos(A)cos(B) − sin (A )sin(B )]

+ i [cos(A)sin(B) + sin(A)cos(B)]

cos(A+B) = cos(A)cos(B) − sin(A )sin(B)

sin (A+B) = sin(A)cos(B) + cos(A)sin (B)

Trigonometric Function (5B) 17 Young Won Lim7/24/14

Sin( angle sum and difference )

sin (A) sin (B)

cos(A) cos(B)

A B

sin (A+B)

sin (A) sin (B)

cos(A) cos(B)

A B

sin (A−B)

sin (A)cos(B) − cos(A)sin (B)

sin (A)cos(B) − cos(A)sin (B)

Trigonometric Function (5B) 18 Young Won Lim7/24/14

Cos( angle sum and difference )

sin (A) sin (B)

cos(A) cos(B)

A B

cos(A+B)

sin (A) sin (B)

cos(A) cos(B)

A B

cos(A−B)

cos(A)cos(B)− sin(A)sin (B)

cos(A)cos(B) + sin(A)sin(B)

Trigonometric Function (5B) 19 Young Won Lim7/24/14

Product to Sum : sin cos

sin(A) sin (B)

cos(A) cos(B)

A B

sin (A+B)

sin (A) sin (B)

cos(A) cos(B)

A B

+sin (A−B)

sin (A+B) + sin (A−B) ⇐ 2sin (A)cos(B)

Trigonometric Function (5B) 20 Young Won Lim7/24/14

Product to Sum : cos sin

sin (A) sin (B)

cos(A) cos(B)

A B

sin (A+B)

sin (A) sin (B)

cos(A) cos(B)

A B

−sin(A−B)

sin (A+B) − sin (A−B) ⇐ 2cos(A)sin (B)

Trigonometric Function (5B) 21 Young Won Lim7/24/14

Product to Sum : cos cos

sin (A) sin (B)

cos(A) cos(B)

A B

cos(A+B)

sin (A) sin (B)

cos(A) cos(B)

A B

+cos(A−B)

cos(A+B) + cos(A−B) ⇐ 2cos(A)cos(B)

Trigonometric Function (5B) 22 Young Won Lim7/24/14

Product to Sum : sin sin

sin (A) sin (B)

cos(A) cos(B)

A B

cos(A+B)

sin (A) sin (B)

cos(A) cos(B)

A B

−cos(A−B)

cos(A+B) − cos(A−B) ⇐ −2sin (A)sin(B)

−cos(A+B) + cos(A−B) ⇐ +2sin (A)sin(B)

Trigonometric Function (5B) 23 Young Won Lim7/24/14

Product to Sum

sin(A) sin (B)

cos(A) cos(B)

A B

sin (A+B)

sin (A) sin (B)

cos(A) cos(B)

A B

+sin (A−B)

sin (A) sin (B)

cos(A) cos(B)

A B

sin (A+B)

sin (A) sin (B)

cos(A) cos(B)

A B

−sin(A−B)

Trigonometric Function (5B) 24 Young Won Lim7/24/14

Product to Sum

sin (A) sin (B)

cos(A) cos(B)

A B

cos(A+B)

sin (A) sin (B)

cos(A) cos(B)

A B

+cos(A−B)

sin (A) sin (B)

cos(A) cos(B)

A B

cos(A+B)

sin (A) sin (B)

cos(A) cos(B)

A B

−cos(A−B)

Trigonometric Function (5B) 25 Young Won Lim7/24/14

Sum and Difference

A

B

A+B = X

A−B = Y

X+Y = A+B+A−B = 2A

X−Y = A+B−A+B = 2B

Trigonometric Function (5B) 26 Young Won Lim7/24/14

Product to Sum

sin ( X+Y2 ) sin ( X−Y

2 )

cos( X+Y2 ) cos( X−Y

2 )

X+Y2

X−Y2

sin (X)

sin ( X+Y2 ) sin ( X−Y

2 )

cos( X+Y2 ) cos( X−Y

2 )

X+Y2

X−Y2

+sin (Y )

sin ( X+Y2 ) sin ( X−Y

2 )

cos( X+Y2 ) cos( X−Y

2 )

X+Y2

X−Y2

sin (X)

sin ( X+Y2 ) sin ( X−Y

2 )

cos( X+Y2 ) cos( X−Y

2 )

X+Y2

X−Y2

−sin(Y )

Trigonometric Function (5B) 27 Young Won Lim7/24/14

Product to Sum

cos(X ) +cos(Y )

cos(X ) −cos(Y )

sin ( X+Y2 )

cos( X+Y2 )

X+Y2

sin ( X+Y2 )

cos( X+Y2 )

X+Y2

sin ( X−Y2 )

cos( X−Y2 )

X−Y2

sin ( X−Y2 )

cos( X−Y2 )

X−Y2

sin ( X+Y2 )

cos( X+Y2 )

X+Y2

sin ( X+Y2 )

cos( X+Y2 )

X+Y2

sin ( X−Y2 )

cos( X−Y2 )

X−Y2

sin ( X−Y2 )

cos( X−Y2 )

X−Y2

Trigonometric Function (5B) 28 Young Won Lim7/24/14

Product-to-Sum & Sum-to-Product

sin (A+B) + sin (A−B) ⇐ 2sin (A)cos(B)

sin (A+B) − sin (A−B) ⇐ 2cos(A)sin (B)

cos(A+B) + cos(A−B) ⇐ 2cos(A)cos(B)

−cos(A+B) + cos(A−B) ⇐ 2sin(A)sin (B)

sin (X ) + sin(Y ) ⇒ 2sin ( X+Y2 )cos( X−Y

2 )

sin (X ) − sin(Y ) ⇒ 2cos( x+Y2 )sin ( X−Y

2 )

cos(X ) + cos(Y ) ⇒ 2cos( X+Y2 )cos( X−Y

2 )

−cos(X ) + cos(Y ) ⇒ 2sin ( X+Y2 )sin ( X−Y

2 )

SUM PRODUCT

SUM PRODUCT

Trigonometric Function (5B) 29 Young Won Lim7/24/14

Derivatives

http://en.wikipedia.org/wiki/Derivative

Trigonometric Function (5B) 30 Young Won Lim7/24/14

Unit Circle Geometry

θ

θ

1 sin (θ)

cos(θ)

θ

1 tan (θ)tan (θ) =sin (θ)

cos (θ)

1

θ =l

2π r2 π =

lr

= l (rad )

r = 1 sin (θ) θ tan (θ)

sin (θ) < θ < tan (θ)

0 < θ < π/2

Trigonometric Function (5B) 31 Young Won Lim7/24/14

Inequalities

θ

θ =l

2π r2 π =

lr

= l (rad )

r = 1 sin (θ) θ tan (θ)

sin (θ) < θ < tan (θ)

0 < θ < π/2

sin (θ)θ

< 1 <tan (θ)

θ

Trigonometric Function (5B) 32 Young Won Lim7/24/14

Sinc(x)

0 < sin (θ) < 10 < sin (θ) < 1

−1 < sin (θ) < 0−1 < sin (θ) < 0

1 < θ

θ < 1

sin (θ)θ

< 1 if θ≠0

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Trigonometric Function (5B) 33 Young Won Lim7/24/14

Sin(x) / x

θ

θ =l

2π r2 π =

lr

= l (rad )

r = 1 sin (θ) θ tan (θ)

sin (θ) < θ < tan (θ)

0 < θ ≪ 1

sin (θ)θ

< 1 <tan (θ)

θ

1 <tan (θ)

θcos(θ) <

sin (θ)θ

cos(θ) <sin (θ)

θ< 1

limθ→0

sin(θ)θ

= 1

Trigonometric Function (5B) 34 Young Won Lim7/24/14

(1 – cos(x)) / x

θ

r = 1

0 < θ < π/2

θ

r = 1

0 < θ ≪ 1

cos(θ) 1−cos(θ)

1−cos(θ)

sin (θ)

θ

θ1−cos(θ)θ

1−cos(θ)θ

1+cos(θ)

1+cos(θ)=

1−cos2(θ)

θ(1+cos(θ))

=sin(θ)

θsin (θ)

1(1+cos(θ))

limθ→0

1−cos(θ)θ

= 0

Trigonometric Function (5B) 35 Young Won Lim7/24/14

sin(x) / x, cos(x) / x, (1 – cos(x)) / x

sin (θ)θ

1−cos(θ)θ

cos(θ)θ

Trigonometric Function (5B) 36 Young Won Lim7/24/14

The Derivative of the Sine Function

dd x

f (x) = limh→0

f (x+h) − f (x)h

dd x

sin(x) = limh→0

sin(x+h) − sin (x)h

= limh→0

sin (x)cos(h) + cos(x)sin (h)− sin(x)h

= limh→0

sin (x)(cos(h) − 1) + cos(x)sin(h)h

= sin(x) limh→0

(cos(h) − 1)h

+ cos(x) limh→0

sin (h)h

= cos(x)

Trigonometric Function (5B) 37 Young Won Lim7/24/14

The Derivative of the Cosine Function

dd x

f (x) = limh→0

f (x+h) − f (x)h

dd x

cos(x) = limh→0

cos(x+h) − cos(x)h

= limh→0

cos(x)cos(h)− sin (x)sin(h) − cos(x)h

= limh→0

cos(x)(cos(h) − 1) − sin (x)sin (h)h

= cos(x)limh→0

(cos(h) − 1)h

− sin(x)limh→0

sin(h)h

= −sin (x)

Trigonometric Function (5B) 38 Young Won Lim7/24/14

The Derivative of the Tangent Function

dd x

f (x) = limh→0

f (x+h) − f (x)h

dd x

tan(x) =dd x ( sin(x)

cos(x) )

=[sin(x)]' cos(x)−sin(x)[cos(x)] '

cos2(x)

=cos2(x)+sin2(x)

cos2(x)

=1

cos2(x)= sec2(x)

Trigonometric Function (5B) 39 Young Won Lim7/24/14

Derivative of sin(x)

f (x) = sin(x)

+1 0 -1 0 +1 0 -1 0 +1 0 -1 0 slope

dd x

f (x ) = cos(x )

leads

Trigonometric Function (5B) 40 Young Won Lim7/24/14

Derivative of cos(x)

f x = cos x

0 -1 0 +1 0 -1 0 +1 0 -1 0 +1 slope

dd x

f x = −sin x

leads

Trigonometric Function (5B) 41 Young Won Lim7/24/14

arcsin(x)

Trigonometric Function (5B) 42 Young Won Lim7/24/14

arccos(x)

Trigonometric Function (5B) 43 Young Won Lim7/24/14

arctan(x)

Trigonometric Function (5B) 44 Young Won Lim7/24/14

Integration

Trigonometric Function (5B) 45 Young Won Lim7/24/14

Inverse Relations

http://en.wikipedia.org/wiki/Derivative

Young Won Lim7/24/14

References

[1] http://en.wikipedia.org/[2] 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”