Trigonometric Identities
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Transcript 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.
RECIPROCAL IDENTITIES
1
cosec
sin
1
sec
cos
1
cot
tan
QUOTIENT IDENTITIES
sin
tan
cos
cot
cos
sin
PYTHAGOREAN IDENTITIES
sin cos 1
2
2
tan 1 sec
2
2
1 cot2 cosec2
EVEN-ODD IDENTITIES
sin sin
cos cos
tan tan
cosec cosec sec sec
cot cot
Establish the following identity: sin cosec
Let's sub in here using reciprocal identity
cos sin
2
2
sin cosec cos sin
1
2
2
sin
cos sin
sin
We are done!
2
We've shown the
LHS equals the
RHS
2
1 cos sin
2
2
sin sin
2
2
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 lefthand 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.
sin
Establish the following identity: cosec cot
Let's sub in here using reciprocal identity and quotient identity 1 cos
sin
We worked on cosec cot
1 cos
LHS and then
RHS but never
1
cos
sin
moved things
sin sin 1 cos FOIL denominator
across the = sign
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
cos sin
1 cos
sin 1 cos
sin
cos 1 cos
11cos
1 cos sin 1 cos
2
sin
1 cos
1 cos sin 1 cos
sin
sin 2
1 cos 1 cos
sin
sin
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
Acknowledgement
I wish to thank Shawna Haider from Salt Lake Community College, Utah
USA for her hard work in creating this PowerPoint.
www.slcc.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 Stephen’s School – Carramar
www.ststephens.wa.edu.au