Download Math 1330 Section 4.4 1 Section 4.4 Trigonometric Expressions and

Math 1330 Section 4.4
Section 4.4
Trigonometric Expressions and Identities
In this section, you’ll learn to simplify trig expressions using identities and using basic algebraic operations.
You can add, subtract, multiply, divide and factor trig expressions, in much the same manner than you can with
algebraic expressions.
Example 1: Perform the following operation and simplify: sec( )  5sec( )  6
Example 2: Factor: sin 2 x  4 sin x  3
Example 3: Factor: cos
Example 4: Simplify:
Example 5: Factor:
1
3
5

cos  sin 
3
10
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Math 1330 Section 4.4
Here are some identities you should already know:
sin(t )
cos(t )
cos(t )
cot(t ) 
sin(t )
tan(t ) 
Reciprocal Identities
1
, sin(t )  0
sin(t )
1
sec(t ) 
, cos(t )  0
cos(t )
csc(t ) 
cot(t ) 
1
, tan(t )  0
tan(t )
Pythagorean Identities
You should either memorize or be able to derive the other two.
sin 2 (t )  cos 2 (t )  1
tan 2 (t )  1  sec 2 (t )
1  cot 2 (t )  csc 2 (t )
We can simplify trigonometric expressions by making substitutions using these identities and performing
appropriate algebra.
Example 6: Simplify: tan  x  csc  x 
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Math 1330 Section 4.4
csc2 ( )  1
Example 7: Simplify:
1  sin 2  
Example 8: Simplify:
sec2  
tan    cot  
At times, you will be asked to verify identities. To do this, you’ll use the identities and algebraic operations to
show that the left-hand side of the problem equals the right-hand side of the problem.
Here are some pointers for helping you verify identities:
1. Remember that your task is to show that the two sides of the equation are equal. You may not assume
that they are equal.
2. Choose one side of the problem to work with and leave the other one alone. You’ll use identities and
algebra to convert one side so that it is identical to the side you left alone. You’ll work with the “ugly”
or more complicated side.
3. If is often helpful to convert all trig functions into sine and cosine. This is usually very helpful! (Unless
it makes things worse!!)
4. Find common denominators, if appropriate.
5. Don’t try to do too much in one step. Take it one step at a time!
6. If working with one side doesn’t get you anywhere, try working with the other side instead.
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Math 1330 Section 4.4
Example 6:
1
 sec2 x
2
sin x  1
Example 7: Prove the identity:
sin x cos x
 cot x
1  cos 2 x
Example 8: Prove the identity:
cot A(1  tan 2 A)
 csc2 A
tan A
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Math 1330 Section 4.4
Example 9: Prove the identity:
sin( x)
 cot( x)  csc( x)
1  cos( x)
You can also use the identities to help you solve problems like this one. (Note, you can also use a triangle to
help you work this problem.)
Example 10: If cot   
5 
,     , find the exact values of the remaining trig functions
12 2
of θ.
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Math 1330 Section 4.4
Quiz help:
1.
2.
3.
4. Simplify:
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Math 1330 Section 4.4
Popper 17 Popper 17 question 1: Simplify. cosx tanx A. sinx B. cosx C. cscx D. secx Popper 17 question 2: Simplify (secx‐tanx)(secx+tanx)
A.
sec2 x+ tan2x B. 1
C. 0
D. 2secx E. None of these
Popper 17 question 3: Simplify cscx‐cosxcotx
A. ‐sinx B. cosxC. tanx D. sinx E. None of these
Popper 17 question 4: Simplify cos3x+sin2xcosx
A. 1
B. secxC. cosx D. tanxE. None of these
Popper 17 question 5: Given tan
E. None of these
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Math 1330 Section 4.4
Solutions to Popper 15:
Given:
4
2
Popper 15 question 1: Find the amplitude.
A. 4 B.
4
C. 2
D. – 2
Popper 15 question 2: Find the horizontal shift (phase shift).
A. Left
B. right 7
C. right
D. left 2
Popper 15 question3 : Find the vertical shift.
A. down 4 B. down
2
C.
down 2
D. down 7
Popper 15 question 4: Write a cos function that has no x-axis reflection, amplitude of 2, horizontal shift of 5 to
the left and period is 2. Place in the form of: f(x)=Acos B(x- C/B) + D
A.
Yes only one answer because I hope this will help.
Popper 15 question 5: Suppose f ( x)  2 csc3x     4 . The phase shift is
A. left
B. right
C. up
D. down
Solutions to Popper 16 on next page:
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M 1330 popper Number 16
1. Simplify
(bubble PS# and popper number)
cos θ
.
sin θ cot θ
D. 1
2. Simplify
(
)
(
)
D. –cot(x)
(
3. Simplify
)
(
)
D. sec( ) + tan ( )
4. Simplify
( )
( )
( )
− .
D. csc( )
5. Simplify
D. sec(x)
(
(
)
)
(
)