Trigonometry Review

Trigonometry ReviewTrigonometry Review

Find Find sinsin((/4) = /4) = coscos((/4) = tan(/4) = tan(/4) =/4) = csc(csc(/4) = sec(/4) = sec(/4) = cot(/4) = cot(/4) =/4) =1

2

22

21

2 2

Evaluate Evaluate tantan((/4)/4)

A.A. Root 2Root 2

B.B. 22

C.C. Root 2 /2Root 2 /2

D.D. 2 / Root 22 / Root 2

E.E. 11

Trigonometry ReviewTrigonometry Review

sin(2sin(2/3) = cos(2/3) = cos(2/3) = /3) = tantan(2(2/3) =/3) = csc(2csc(2/3) = /3) = secsec(2(2/3) = cot(2/3) = cot(2/3) =/3) =

31

322

3

1

2

3

2

Evaluate Evaluate secsec(2(2/3)/3)

A.A. -1-1

B.B. -2-2

C.C. -3-3

D.D. Root(3)Root(3)

E.E. 2 / Root(3)2 / Root(3)

Trig. DerivativesTrig. Derivatives

sin’(x) = cos(x) cos’(x) = - sin’(x) = cos(x) cos’(x) = - sin(x)sin(x)


sin’(x) = cos(x)sin’(x) = cos(x)

sin’(x) = sin’(x) = 0

sin( ) sin( )limh

x h x

h

sin’(x) = .sin’(x) = .

sin’(x) = sin’(x) =


0

sin( ) sin( )limh

x h x

h

0

sin( )cos( ) sin( ) cos( ) sin( )limh

h x x h x

h

0

sin( )cos( ) sin( )[cos( ) 1]limh

h x x h

h

0 0

sin( ) cos( ) sin( )[cos( ) 1]lim limh h

h x x h

h h

Rule 4 says .Rule 4 says .

A.A. 00

B.B. 0.50.5

C.C. 11

D.D. 1.51.5

0

sin( )limh

h

h

Rule 5 says .Rule 5 says .

A.A. 00

B.B. 0.50.5

C.C. 11

D.D. 1.51.5

0

cos( ) 1limh

h

h

sin’(x) = sin’(x) = . .



1*cos( ) sin( )*0x x

cos( )x

0 0

sin( ) cos( ) sin( )[cos( ) 1]lim limh h

h x x h

h h


sin’(x) = cos(x) cos’(x) = - sin’(x) = cos(x) cos’(x) = - sin(x)sin(x)

If y = sin(x) + 2xIf y = sin(x) + 2x22, find , find dy/dxdy/dx

A.A. - cos(x) + 4x- cos(x) + 4x

B.B. cos(x) + 4cos(x) + 4

C.C. cos(x) + 4xcos(x) + 4x


sinsin’(x) = ’(x) = coscos(x) cos’(x) = - sin(x)(x) cos’(x) = - sin(x)

A) A) sinsin’(0) = ’(0) = coscos(0) = 1(0) = 1 B) B) sinsin’(’(/4) = /4) = coscos((/4) = 0.707/4) = 0.707 C) C) sinsin’(-’(-/3) = /3) = coscos(-(-/3) = 0.5/3) = 0.5

x= 0, 2x= 0, 2/3, - /3, - 33/4/4

coscos’(x) = ’(x) = - sinsin(x)(x) A) A) coscos’(0) = ’(0) = -- sinsin (0) = 0 (0) = 0 B) B) coscos’(-3’(-3/4) = /4) = -- sinsin(5(5/4) = 0.707/4) = 0.707 C) C) coscos’(2’(2/3) = /3) = -- sinsin(2(2/3) = - 0.866/3) = - 0.866

Evaluate cos’(Evaluate cos’(/2)/2)

A.A. -1-1

B.B. -.707-.707

C.C. 11

D.D. 0.7070.707

Evaluate sin’(Evaluate sin’(/3)/3)

A.A. - 0.5- 0.5

B.B. 0.50.5

C.C. 0.7070.707

D.D. 0.8660.866


sin’(x) = cos(x) cos’(x) = - sin(x)sin’(x) = cos(x) cos’(x) = - sin(x) tan’(x) = sectan’(x) = sec22(x) cot’(x) = - csc(x) cot’(x) = - csc22(x) (x) sec’(x) = sec(x)tan(x) csc’(x) = -sec’(x) = sec(x)tan(x) csc’(x) = -

csc(x)cot(x) csc(x)cot(x)

Trig. Trig. DerivativesDerivatives

Theorem tan’(x) = secTheorem tan’(x) = sec22(x)(x) Proof : tan’(x) = [sin(x)/cos(x)]’ Proof : tan’(x) = [sin(x)/cos(x)]’

2

cos( )cos( ) sin( )[ sin( )]

cos ( )

x x x x

x

2

1

cos ( )x


Theorem tan’(x) = secTheorem tan’(x) = sec22(x)(x) tan’(tan’(/4) =/4) =

2sec ( )4

1

cos( )4 2

22 2


Theorem tan’(x) = secTheorem tan’(x) = sec22(x)(x) tan’(tan’(/4) = sec/4) = sec22((/4) = 2 while tan(/4) = 2 while tan(/4) = /4) = 11


Theorem cot’(x) = - cscTheorem cot’(x) = - csc22(x)(x) Proof : cot’(x) = [cos(x)/sin(x)]’ Proof : cot’(x) = [cos(x)/sin(x)]’

2

sin( )[ sin( )] cos( )cos( )

sin ( )

x x x x

x

2

1

sin ( )x


Theorem sec’(x) = sec(x)tan(x)Theorem sec’(x) = sec(x)tan(x) Proof : sec’(x) = [1/cos(x)]’ Proof : sec’(x) = [1/cos(x)]’

2

cos( )0 1( sin( )

cos ( )

x x

x

sin( ) 1

cos( ) cos( )

x

x x


Theorem csc’(x) = - csc(x)cot(x)Theorem csc’(x) = - csc(x)cot(x) Proof : csc’(x) = [1/sin(x)]’ Proof : csc’(x) = [1/sin(x)]’

2

sin( )0 1cos( )

sin ( )

x x

x

cos( ) 1

sin( ) sin( )

x

x x


sin’(x) = cos(x) cos’(x) = - sin(x)sin’(x) = cos(x) cos’(x) = - sin(x) tan’(x) = sectan’(x) = sec22(x) cot’(x) = - csc(x) cot’(x) = - csc22(x) (x) sec’(x) = sec(x)tan(x) csc’(x) = - sec’(x) = sec(x)tan(x) csc’(x) = -

csc(x)cot(x) csc(x)cot(x)

If y = tan(x) sec(x) find theIf y = tan(x) sec(x) find thevelocity and y’(velocity and y’(/3)/3)

sec’(x) = sec(x)tan(x) tan’(x) = secsec’(x) = sec(x)tan(x) tan’(x) = sec22(x)(x)

y ’ = tan(x)sec(x)tan(x) + sec(x)secy ’ = tan(x)sec(x)tan(x) + sec(x)sec22(x) (x)

y’=sec(x)[secy’=sec(x)[sec2 2 (x)-1] + sec(x)-1] + sec33(x)=2sec(x)=2sec33(x)-(x)-sec(x)sec(x)

y’(y’(/3) = 2sec/3) = 2sec33((/3)-sec(/3)-sec(/3) =/3) =

sinsin22x+cosx+cos22x=1 dividing by cosx=1 dividing by cos2(2(x)x)

tantan2 2 (x)+1=sec(x)+1=sec2 2 (x)(x)

32 2 2 14

If y = tan(x) cos(x) find theIf y = tan(x) cos(x) find theacceleration and y’’(acceleration and y’’(/3)/3)

y’ = cos(x)y’ = cos(x)

y’’ = -sin(x) y’’(y’’ = -sin(x) y’’(/3)=/3)=3

2

If y = tan(x) + cos(x) find theIf y = tan(x) + cos(x) find theinitial acceleration, y’’(0)initial acceleration, y’’(0)

tan’(x) = sectan’(x) = sec22(x) sec’(x) = (x) sec’(x) = sec(x)tan(x)sec(x)tan(x)

y’ = sec(x)sec(x) - sin(x) y’’ = y’ = sec(x)sec(x) - sin(x) y’’ =

sec(x) sec(x)tan(x) + sec(x) sec(x)tan(x) - sec(x) sec(x)tan(x) + sec(x) sec(x)tan(x) - cos(x)cos(x)

= 2 sec= 2 sec22(x) tan(x) – cos(x) (x) tan(x) – cos(x)

y’’(0) = 2 * 1 * 0 - . . . . . . y’’(0) = 2 * 1 * 0 - . . . . . .

y” = 2 secy” = 2 sec22(x) tan(x) – cos(x)(x) tan(x) – cos(x)y”(0) =y”(0) =

If y = sec(x), find the acceleration, If y = sec(x), find the acceleration, y’’(0) using the product rule on y’’(0) using the product rule on

sec’(x).sec’(x).

Find the slope of the tangent Find the slope of the tangent line to y = x + sin(x) when x = line to y = x + sin(x) when x =

00

Write the equation of the line Write the equation of the line tangent to y = x + sin(x) when x tangent to y = x + sin(x) when x

= 0= 0A.A. y = 2x + 1y = 2x + 1

B.B. y = 2x + 0.5y = 2x + 0.5

C.C. y = 2xy = 2x

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Trigonometry Review