This is a movie about Trigonometry

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This is a movie aboutTrigonometry

• C3 Reciprocal and Inverse Trig Functions
• Directed by J Wathall and her Year13 A level
  Maths class

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Reciprocal functions

• What is the reciprocal of y 3x 3 ?
• Yes it is
• The reciprocal means ONE OVER the function. Or in
  a fraction it means to change the denominator and
  numerator

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Inverse functions

• The inverse of a function maps the output of a
  function back to the input. THIS IS NOT THE
  RECIPROCAL!
• For example the function y 3x 3 has an
  inverse of
• Notice the inverse is not the same as the
  reciprocal. The inverse is NOT one over!

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Reciprocal Trig Functions

• What is the reciprocal of cos x?
• What is the reciprocal of sin x?
• What is the reciprocal of tan x?
• We have special names for these reciprocal
  functions.

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Here they are
Here we must remember that the denominator cannot
equal zero so cos x, sin x and tan x are not
defined for the value zero.
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Example 1

• Volunteer Using your calculator evaluate sec
  1000 , cosec 2600 and
• cot( 4?/3) c to 3 sig figs.
• Volunteer WAC evaluate the exact value of cot
  1350, sec 2250 and cot( 4?/3) c

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What do the reciprocal graphs look like?

• 1) Complete this table for y sec x
• 2) Sketch the curve y cos x
• for -180 lt x lt 180
• 3) Using a different coloured pen now
  sketch y sec x

x 0 30 45 60 70 80 85 95 100 110 120 135 150 180 210
y
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A review of last lesson

• Do you remember how to sketch the reciprocal trig
  functions?
• Sketch y cos x and on the same curve sketch y
  sec x for -180ltxlt180 labeling all asymptotes

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Tada!
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Or on a larger scale y secx
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Facts about y sec x

• Write down when the asymptotes occur.
• X 900, 2700 etc
• What is the period of the curve? (one full cycle)
• 3600

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What is the difference between the graphs of y
sinx and y cos x?

• Yes you are correct.
• So the y cosec x curve is exactly the same as
  the y sec x curve but a shift to the right by
  90 0.
• Can you sketch this on your graph paper using
  another colour. Dont forget to draw your
  asymptotes

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Y cosec x- blue curve
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Facts about y cosec x

• Write down when the asymptotes occur.
• X 1800, 3600, etc
• What is the period of the curve? (one full cycle)
• 3600

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Lastly y cot x Write down three facts about
this curve.
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Y cot x

• Write down when the asymptotes occur.
• X0,1800, etc
• What is the period of the curve? (one full cycle)
• 1800

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Transformations of the Reciprocal Trig Functions.

• Let us use Autograph to help us understand these
  transformations.
• See worksheet work through guided examples.
• Homework Monday 27th Aug
• If you want an A All of ex 6A, 6B
• If you want a B every other question in 6A,6B for
  Wednesday

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Simplifying Trig expressions

• Examples Simppppplify
• Sinxsecx
• Sinxcosx(secxcosecx)

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Showing volunteer

• Cotx cosecx cos3 x
• Sec2 x cosec2x
• Q 1,2,3 and 4 Ex 6C

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Showing Melody

• Cotx cosecx cos3 x
• Sec2 x cosec2x

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Q4f Show that
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Homework help!

• Is this a quadratic? Ex 6H

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Showing

• Cotx cosecx cos3 x
• Sec2 x cosec2x

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Ex 6c q6H

• A quadratic in disguise

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Solving trig equations

• Sec x -2.5 for the interval 0ltxlt360
• Cot 2x 0.6 for the interval 0ltxlt360
• Ex 6C 5,6,7

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Solving Gillean, Jocelyn

• Secx -2.5
• Cot 2x 0.6

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Homework 6C
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6C q7D
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Another form of an Identity

• Starting with the identity
• Divide this equation by cos 2x.
• Divide this equation by sin2 x.

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Two new identities
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Lots of examples

• If tan x -5/12 and x is obtuse find the exact
  value of
• A) sec x
• B) sin x
• Use a RAT

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More examples

• Prove

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One more interesting one
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Ex 6D more practice
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The Inverse Trig Functions

• Remember an inverse means a function which maps
  the output back to the input and the graph is a
  reflection about the line y x.
• So we do not confuse the reciprocal trig
  functions we use a special notation for the
  inverse trig functions.
• The are called arcsinx, arccosx and arctanx.

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Some conditions

• For an inverse function to exist the function
  must be a one to one mapping. We restrict the
  domain of y sin x, y cos x and y
  tan x for the inverse to exist.
• Let us use Autograph again to help us see what
  arcsinx, arccosx and arctanx looks like.

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One to one mapping y cos x
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y arccosx
Here the domain is -1ltxlt1
The range is 0ltylt ?
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Yarccosx

• You must remember here that the domain is
  restricted to
• 0 x ?
• So if we were simplifying
• We would only look at the second quadrant
• Why?

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Example

• Simplify the following
• This is the same as

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Y sin x

• Go to www.mathsnet.net for beautiful applet

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Y arcsin x
Domain -1ltxlt1
Range -?/2ltylt ?/2
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Domain

• Here for y arcsinx the domain is
• -?/2 x ?/2
• So to simplify a problem like this
• We only look at fourth quadrant why?

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Y arctan x
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Domain of y arctanx

• You can see x is real so the domain is
• The range is
• So simplifying
• We find

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Inverse trig applets

• Click here
• Inverse trig graphs as a reflection

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Example

• Click here for worked examples

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Ex 6E

• Q6b

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Mixed exercise 6F

• Proving identities

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Using trig identities

• Solve the equation 4cosec2 x -9 cot x for 0 x
  360

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The End

• The mind map
• Click here