Download chapter 2 analytical trigonometry

CHAPTER 2.1
CHAPTER 2 ANALYTICAL TRIGONOMETRY
PART 1 – Using Fundamental Identities
TRIGONOMETRY MATHEMATICS CONTENT STANDARDS:
2
2
 3.0 - Students know the identity cos (x) + sin (x) = 1:
3.1 - Students prove that this identity is equivalent to the Pythagorean
theorem (i.e., students can prove this identity by using the
Pythagorean Theorem and, conversely, they can prove the
Pythagorean Theorem as a consequence of this identity).


3.2 - Students prove other trigonometric identities and simplify others by
2
2
using the identity cos (x) + sin (x) = 1. For example, students use
2
2
this identity to prove that sec (x) = tan (x) + 1.
5.0 – Students know the definitions of the tangent and cotangent functions and
can graph them.
6.0 – Students know the definitions of secant and cosecant functions and
can graph them.
OBJECTIVE(S):
 Students will learn how to evaluate trigonometric functions.
 Students will learn how to simplify trigonometric expressions.
 Students will learn how to develop additional trigonometric identities.
 Students will learn how to solve trigonometric equations.
Fundamental Trigonometric Identities
Reciprocal Identities
=
cos u =
tan u =
csc u =
sec u =
cot u
=
cot u
=
sin u
Quotient Identities
tan u =
CHAPTER 2.1
Cofunction Identities


sin   u 
=
2



cos  u 
2

=


tan   u 
2

=


cot   u 
2

=


sec  u 
2

=


csc  u 
2

=
Negative Angle Identities
=
sin  u 
cos u 
=
tan  u 
=
cot  u 
=
sec u 
=
csc u 
=
Pythagorean Identities
1.
2.
3.
Pythagorean identities are sometimes use in radical form such as
sin u
=
tan u =
where the sign depends on the choice of u.
Using the Fundamental Identities
One common use of trigonometric identities is to use given values of trigonometric
functions to evaluate other trigonometric functions.
CHAPTER 2.1
EXAMPLE 1: Using Identities to Evaluate a Function
3
Use the values of sec u   and tan u  0 to find the values of all six trigonometric
2
functions.
Using a reciprocal identity, you have
cos u =
=
=
Using a Pythagorean identity, you have
sin 2 u = 1 cos 2 u
Pythagorean identity.
=
Substitute _____ for cos u .
=
Simplify.
=
Simplify.
Because sec u  0 and tan u  0 , it follows that u lies in Quadrant ______. Moreover,
because sin u is negative when u is in Quadrant _____, you can choose the negative root
and obtain
sin u = _______. Now, knowing the values of the sine and cosine, you can find the
values of all six trigonometric functions.
sin u
=
csc u =
=
=
cos u =
sec u =
=
tan u =
cot u
=
=
=
=
=
CHAPTER 2.1
EXAMPLE 2: Simplifying a Trigonometric Expression
Simplify sin x cos 2 x  sin x .
First factor out a common monomial factor and then use a fundamental identity.
sin x cos 2 x  sin x
=
Factor out monomial
factor.
=
Factor out _____.
=
Pythagorean identity.
=
Multiply.
When factoring trigonometric expressions, it is helpful to find a special polynomial
factoring form that fits the expression.
EXAMPLE 3: Factoring Trigonometric Expressions
Factor each expression.
a.) sec 2   1
Here the expression has the polynomial form u 2  v 2 (the difference of two squares),
which factors as
sec 2   1
=
b.) 4 tan 2   tan   3
This expression has the polynomial form, ax 2  bx  c , and it factors as
4 tan 2   tan   3
=
On occasion, factoring or simplifying can best be done by first rewriting the expression in
terms of just one trigonometric function or in terms of sine or cosine only.
CHAPTER 2.1
EXAMPLE 4: Factoring a Trigonometric Expression
Factor csc 2 x  cot x  3 .
You can use the identity ______________________to rewrite the expression in terms of
the cotangent alone.
csc 2 x  cot x  3
=
Pythagorean identity.
=
Combine Like Terms.
=
Factor.
EXAMPLE 5: Simplifying a Trigonometric Expression
Simplify sin t  cot t cos t .
Begin by rewriting the expression in terms of sine and cosine.
sin t  cot t cos t
=
Quotient identity.
=
Add fractions.
=
Pythagorean identity.
=
Reciprocal identity.
CHAPTER 2.1
EXAMPLE 6: Adding Trigonometric Expressions
Perform the addition and simplify.
sin 
cos 

1  cos  sin 
sin 
cos 

1  cos  sin 
=
=
=
Multiply.
=
Pythagorean
identity: _________
=
Divide out common
factor.
=
Reciprocal identity.
EXAMPLE 7: Rewriting a Trigonometric Expression
1
Rewrite
so that it is not in fractional form.
1  sin x
From the Pythagorean identity cos 2 x  1  sin 2 x  1  sin x 1  sin x  , you can see that by
multiplying both the numerator and the denominator by _______________ you will
produce a monomial denominator.
1
1  sin x
=
Multiply numerator and denominator
by ___________.
=
Multiply.
=
Pythagorean identity.
=
Write as separate fractions.
CHAPTER 2.1
=
Product of fractions.
=
Reciprocal and quotient identities.
DAY 1
EXAMPLE 8: Trigonometric Substitution

Use the substitution x  2 tan  , 0    , to write
2
of  .
4  x 2 as a trigonometric function
Begin by letting x  2 tan  . Then, you can obtain
4  x2
=
Substitute _________ for x.
=
Rule of exponents.
=
Factor.
=
Pythagorean identity.
=
sec   0 for 0   
You can use a triangle to check your solution.
x  2 tan 
x
2
Angle whose tangent is ______

2
.
CHAPTER 2.1
1.) Simplify the trigonometric expressions.
a. tan x  cos x


d. 1  sin 2 x sec x


sin   x 
2

b.


cos  x 
2


c. cot x  sec x

e. sin 2 x csc 2 x  1


cos 2   x 
2

f.
cos x
CHAPTER 2.1
2.) Simplify.
a. 1  2 sin 2 x  sin 4 x
c. tan x 
DAY 2
sec 2 x
tan x
b. cot x  csc xcot x  csc x
d. ln csc  ln tan 