You need to memorize the fundamental trigonometric identities on page 532 in
your ... o Use algebra and fundamental identities to simplify the expression.
Simplifying Trigonometric Expressions Objective: To use algebra and fundamental identities to simplify a trigonometric expression β’ β’
β’
You need to memorize the fundamental trigonometric identities on page 532 in your textbook.
You need to be able to recognize rearrangements of fundamental identities. In particular, you often see rearrangements of Pythagorean Identities. For example, sin2 π₯ + cos2 π₯ = 1 βΉ sin2 π₯ = 1 β cos2 π₯ sin2 π₯ + cos2 π₯ = 1 βΉ cos 2 π₯ = 1 β sin2 π₯ Simplifying trigonometric expressions often takes some trial and error, but the following strategies may be helpful. o o o o o
Strategy
Use algebra and fundamental identities to simplify the expression. Sometimes, writing all functions in terms of sines and cosines may help. Sometimes, combining fractions by getting a common denominator may help. π+π π π Sometimes, breaking one fraction into two fractions may help: = + π
Sometimes, factoring may help.
Rewriting in terms of sine and cosine
Example
π
π
Approach sin π₯ tan π₯ cos π₯ = 1 sec π₯ cos π₯ =
sin π₯ cos π₯ β cos π₯ 1
= sin π₯
β’ β’ β’
β’
sin π₯
tan π₯ = cos π₯ 1
sec π₯ = cos π₯
To divide by a fraction, multiply by the reciprocal of the denominator Reduce the resulting product
Factor out a common factor of cos π₯ Use the identity: cos 2 π₯ = 1 β sin2 π₯ Use a property of exponents to multiply cos π₯ and cos 2 π₯ cot π₯ =
cos π₯ sin π₯
Get a common denominator of sin π₯ and add the two fractions sin2 π₯ + cos 2 π₯ = 1 1
csc π₯ = sin π₯ π+π π
π
π
=π+π
sec π₯ divided by itself is 1 1
sec π₯ = cos π₯ so cos π₯ sec π₯
= cos 2 π₯
1 β cos 2 π₯ = sin2 π₯
Simplifying Trigonometric Expressions Simplify the following expressions. 1) sin π₯ cot π₯ 2) 3)
sec π₯ csc π₯
1βsin2 π₯ cos π₯
4) sin π‘ β sin π‘ cos2 π‘ 5) cos π₯ + tan π₯ sin π₯ 6) sin3 π₯ + sin π₯ cos2 π₯ 7) 8)
csc π₯βsin π₯ csc π₯
sin π₯
cos π₯
+
cos π₯
1+sin π₯
Simplifying Trigonometric Expressions Solutions: 1) sin π₯ cot π₯ = sin π₯ β 2) 3)
sec π₯ csc π₯
=
1βsin2 π₯ cos π₯
1 cos π₯ 1 sin π₯
=
=
1
cos π₯
cos2 π₯ cos π₯
β
cos π₯ sin π₯
sin π₯ 1
= cos π₯ =
sin π₯
cos π₯
= tan π₯
= cos π₯
4) sin π‘ β sin π‘ cos2 π‘ = sin π‘ β (1 β cos 2 π‘ ) = sin π‘ β sin2 π‘ = sin3 π‘ 5) cos π₯ + tan π₯ sin π₯ = cos π₯ +
sin π₯
cos π₯
β sin π₯ =
cos2 π₯+sin2 π₯ cos π₯
=
1
cos π₯
= sec π₯
6) sin3 π₯ + sin π₯ cos2 π₯ = sin π₯ β (sin2 π₯ + cos2 π₯ ) = sin π₯ 7) 8)
csc π₯βsin π₯ csc π₯
sin π₯
cos π₯
+
=
cos π₯
csc π₯ csc π₯
1+sin π₯
=
β
sin π₯
csc π₯
= 1 β sin2 π₯ = cos2 π₯
sin π₯β(1+sin π₯)+cos π₯βcos π₯ cos π₯β(1+sin π₯)
=
sin π₯+sin2 π₯+cos2 π₯ cos π₯β(1+sin π₯)