TRIGONOMETRIC IDENTITIES

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TRIGONOMETRIC IDENTITIES
Remember an identity is an equation that is true
for all defined values of a variable.
We are going to use the identities that we have
already established to "prove" or establish other
identities. Let's summarize the basic identities
we have.
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Establish the following identity
Let's sub in here using reciprocal identity
We are done! We've shown the LHS equals the RHS
We often use the Pythagorean Identities solved
for either sin2? or cos2?. sin2? cos2? 1
solved for sin2? is sin2? 1 - cos2? which is
our left-hand side so we can substitute.
In establishing an identity you should NOT move
things from one side of the equal sign to the
other. Instead substitute using identities you
know and simplifying on one side or the other
side or both until both sides match.
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Establish the following identity
Let's sub in here using reciprocal identity and
quotient identity
We worked on LHS and then RHS but never moved
things across the sign
FOIL denominator
combine fractions
Another trick if the denominator is two terms
with one term a 1 and the other a sine or cosine,
multiply top and bottom of the fraction by the
conjugate and then you'll be able to use the
Pythagorean Identity on the bottom
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Hints for Establishing Identities

• Get common denominators

• If you have squared functions look for
  Pythagorean Identities

• Work on the more complex side first

• If you have a denominator of 1 trig function
  try multiplying top bottom by conjugate and use
  Pythagorean Identity

• When all else fails write everything in terms of
  sines and cosines using reciprocal and quotient
  identities

• Have fun with these---it's like a puzzle, can you
  use identities and algebra to get them to match!

MathXTC ?
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Acknowledgement I wish to thank Shawna Haider
from Salt Lake Community College, Utah USA for
her hard work in creating this PowerPoint. www.sl
cc.edu Shawna has kindly given permission for
this resource to be downloaded from
www.mathxtc.com and for it to be modified to suit
the Western Australian Mathematics Curriculum.
Stephen Corcoran Head of Mathematics St
Stephens School Carramar www.ststephens.wa.edu.
au