pre calculus problems of the day simplify the following:
TRANSCRIPT
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Pre calculus Problems of the Day
Simplify the following:
a) sin6
2
cos6
2
b) sin4
2
cos4
2
c) sin43
2
cos43
2
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Trigonometric Identities - a statement of equality that is true for all values where the function is defined.
Reciprocal Identities:
sin 1csc
cos 1sec
tan 1cot
csc 1sin
sec 1cos
cot 1tan
Quotient Identities:
tan sincos
cot cossin
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Pythagorean Identities:
sin2 cos2 1
1cot2 csc2
tan2 1sec2
Even/Odd Identities:
sin sin
cos cos
tan tan
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Verifying Trigonometric Identities
To verify a trigonometric identity we must show that one side of the identity can be simplified so that it is identical to the other side or each side can be simplified independently until they are identical.
Never treat an identity like an equation. We are not solving for the variable.
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WE VERIFY (OR PROVE) IDENTITIES BY DOING THE FOLLOWING:
• Work with one side at a time.• We want both sides to be exactly the same.• Start with the more complicated side• Use algebraic manipulations and/or the basic
trigonometric identities until you have the same expression as on the other side.
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Techniques for verifying trigonometric identities.
1) Rename using the fundamental identities.
2) Rewrite a more complicated side in terms of sines and cosines.
3) Factor.
4) Combine fractional expressions using an LCD.
5) Separate a single-term quotient into two terms.
6) Multiply the numerator and denominator on one side by a binomial factor that appears on the other side of the identity.
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Verifying Trigonometric Identities
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2 2 2tan x cos x 1 cos x 2
2 22
sin xcos x 1 cos x
cos x
2 2sin x 1 cos x 2 21 cos x 1 cos x
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tan csc sec
sin 1sec
cos sin
1sec
cos
sec sec
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sinx sec x cot x 1
1 cosxsinx 1
cosx sinx
1 1
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tanA cot A csc Asec A
sin A cos Acsc A sec A
cos A s
sin A cos A
sin A cosn A Ai
2 2sin A cos Acsc A sec A
sin A cos A
1csc A sec A
sin A cos A
1 1csc A sec A
sin A cos A
csc A sec A csc A sec A
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csc xcos x
cot x tanx
1
sinx cos xcos x sinxsinx cos x
2 2
1sinx cos x
cos x sin xsinx cos x
2 2
1 sinx cos xcos x
sinx cos x sin x
1 sinx cos xcos x
sinx 1
cos x cos x
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21 12sec y
siny 1 siny 1
21 12sec y
siny 1 sin
siny 1
y 1
siny 1
siny 1 siny 1
2siny 1 siny 12sec y
siny 1 siny 1
2
2
22sec y
sin y 1
2
2
22sec y
1 sin y 1
22
22sec y
cos y
2 22sec y 2sec y
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4 2 4cos x 1 2sin x sin x
22 2 4cos x 1 2sin x sin x
22 2 41 sin x 1 2sin x sin x
2 2 2 41 sin x 1 sin x 1 2sin x sin x
2 4 2 41 2sin x sin x 1 2sin x sin x
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cos xtanx sec x
1 sinx
sinx cos x
sec xcos x 1 sinx
sinx 1 sinx cos x cos xsec x
cos x 1 sinx
2 2sinx sin x cos x
sec xcos x 1 sinx
sinx 1
sec xcos x 1 sinx
1
sec xcos x
sec x sec x
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2
sinx 1csc x
1 cos x sinx
2
sinx 1csc x
sin x sinx
sinx 1
csc xsinx 1 sinx
1csc x
sinx
csc x csc x
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2cot x cot x tanx csc x
2 2cot x 1 csc x 2 2csc x csc x
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cot x csc xcot x csc x
sinx tanx
cos x 1sinx sinx cot x csc x
sinxsinx
cos x
cos x 1sinx sinx cot x csc x
sinx cos x sinxcos x cos x
cos x 1 cos x
cot x csc xsinx sinx cos x 1
1 cos xcot x csc x
sinx sinx
cot x csc x cot x csc x