Simplifying Trigonometric Expressions
Objective:
To use algebra and fundamental identities to simplify a trigonometric expression
• 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,
sin
+ cos
= 1 sin
= 1 cos
sin
+ cos
= 1 cos
= 1 sin
• Simplifying trigonometric expressions often takes some trial and error, but the following strategies may be helpful.
o Use algebra and fundamental identities to simplify the expression.
o Sometimes, writing all functions in terms of sines and cosines may help.
o Sometimes, combining fractions by getting a common denominator may help.
o Sometimes, breaking one fraction into two fractions may help:
=
+
o
Sometimes, factoring may help.
Strategy Example Approach
Rewriting in terms of
sine and cosine
tan
sec
=
sin
cos
1
cos
=
sin
cos
cos
1
= sin
•
tan =
• sec =
• To divide by a fraction,
multiply by the reciprocal
of the denominator
• Reduce the resulting
product
Simplifying Trigonometric Expressions
Strategy Example Approach
Factoring
cos cos sin
= cos
(
1 sin
)
= cos cos
= cos
•
Factor out a common
factor of cos
• Use the identity:
cos
= 1 sin
• Use a property of
exponents to multiply
cos and cos
Getting a common
denominator
sin + cos cot = sin + cos
cos
sin
=
sin
sin
+
cos
sin
=
sin
+ cos
sin
=
1
sin
= csc
•
cot =
• Get a common
denominator of sin and
add the two fractions
• sin
+ cos
= 1
• csc =
Splitting one fraction
into two fractions
sec cos
sec
=
sec
sec
cos
sec
= 1 cos
= sin
•
=
+
• sec divided by itself is 1
• sec =
so
= cos
•
1 cos
= sin
Simplifying Trigonometric Expressions
Simplify the following expressions.
1) sin cot
2)
3)
4) sin sin cos
5) cos + tan sin
6) sin
+ sin cos
7)
8)
+
Simplifying Trigonometric Expressions
Solutions:
1) sin cot = sin
= cos
2)
=
=
=
= tan
3)
=
= cos
4) sin sin cos
= sin
(
1 cos
)
= sin sin
= sin
5) cos + tan sin = cos +
sin =
=
= sec
6) sin
+ sin cos
= sin
(
sin
+ cos
)
= sin
7)
=
= 1 sin
= cos
8)
+
=
( )
( )
=
( )
=
( )
=
= sec
