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Verifying Trig Identities (5.1). JMerrill, 2009 (contributions from DDillon). Trig Identities. Identity : an equation that is true for all values of the variable for which the expressions are defined Ex: or (x + 2) = x + 2 - PowerPoint PPT Presentation
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Verifying Trig Verifying Trig IdentitiesIdentities
(5.1)(5.1)JMerrill, 2009JMerrill, 2009
(contributions from DDillon)(contributions from DDillon)
Trig IdentitiesIdentity: an equation that is true
for all values of the variable for which the expressions are defined
Ex: or (x + 2) = x + 2Conditional Equation: only true for
some of the valuesEx: tan x = 0 or x2 + 3x + 2 = 0
sinxtanx cosx
Recallsin csc
cos sec
tan cot
y rr yx rr xy xx y
Recall - IdentitiesReciprocal Identities
1cscsin
1seccos
1cottan
Also true:
1sincsc
1cossec
1tancot
Recall - IdentitiesQuotient Identities
sintancos
coscotsin
Fundamental Trigonometric
IdentitiesNegative Identities (even/odd)
sin sin csc csc
tan tan cot cot
cos cos sec sec
These are the only even functions!
Recall - IdentitiesCofunction Identities
sin cos2
cos sin
2
tan cot2
cot tan
2
sec csc2
csc sec2
Recall - IdentitiesPythagorean Identities
2 2sin cos 1
2 2tan 1 sec
2 21 cot csc
Simplifying Trig Expressions
• Strategies• Change all functions to sine and cosine
(or at least into the same function)• Substitute using Pythagorean Identities• Combine terms into a single fraction with
a common denominator• Split up one term into 2 fractions• Multiply by a trig expression equal to 1• Factor out a common factor
Simplifying # 1cot sinx x
cos sinsin
x xx
cossin
xx
sin x
cos x
Simplifying #22cos sin
sinx xx
2 2cos sinsin sin
x xx x
2 2cos sinsin
x xx
1sin
x
csc x
Simplifying #32
2
1 coscos x
x2
2
sincos
xx2tan x
Simplifying #4cos sin tanx x x
sincos sincos
xx xx
2sincoscos
xxx
2 2cos sincos cos
x xx x
1cos
x
sec x
Simplifying #53 2sin cos cos
2
x x x
3 2sin sin cos x x x
2 2sin sin cos x x x
sin x
Proof Strategies• Never cross over the equal sign (you
cannot assume equality)• Transform the more complicated side of
the identity into the simpler side.• Substitute using Pythagorean identities.• Look for opportunities to factor• Combine terms into a single fraction
with a common denominator, or split up a single term into 2 different fractions
• Multiply by a trig expression equal to 1.• Change all functions to sines and
cosines, if the above ideas don’t work.ALWAYS TRY SOMETHING!!!
Example• Prove• 2 fractions that need to be added:• Shortcut:
sin cos csc1 cos sin
sin sin cos 1 cos1 cos sin
2 2sin cos cos1 cos sin
1 cos
1 cos sin
1 cscsin
2 2 2 cos 1 cot cot Show x x x
2 2cos 1 cotx x
2 2cos cscx x
22
1cossinx
x2
2
cossin
xx
2cot x
1 + cot2x = csc2 x
22
1cscsin
xx
2
tan cot Prove tancsc
x x xx
2
tan cotcsc
x xx
2
sin coscos sin
csc
x xx x
x
2 2
2
sin cossin cos
csc
x xx xx
2
1sin cos
1sin
x x
x21 sin
sin cos 1
xx x
sincosxx
tan x2
sin coscos sin
csc
x xx x
x