Trig Unit 5 Packet

Trigonometry
Unit 5 Packet
Name_____________________ Hr___
Mathematical Identities
An identity is an equation that is true for ALL possible values of the value for which the
equation is defined.
EXAMPLE:
Most equations are not identities because an equation is not necessarily true for all values
of the variables involved. Circle the equations that are IDENTITIES:
2  x  x2  x  13
x2  y 2  x  y
x 6  x3
 x  7
2
 x 2  49
 x  5 x  5  x2  25
x2  x
The most commonly used identities in Trigonometry are the Pythagorean identities. Why
do you think they are called by that name?
1. Use the Pythagorean Thm to write an equation that relates x, y, and r.
2. What ratio is equal to
?
3. What ratio is equal to
?
4. Using substitution and simplification, combine the three equations above to write a
single equation in terms of θ.
Since the identity should be true for ALL values, let’s explore angles in all four quadrants:
QI
QII
QIII
QIV
Trigonometry
Unit 5 Packet
Trigonometry Triangles
Directions:

Place the names of the six trigonometric functions in the order which we learned
them (
at the vertices labeled A, B, C, D, E,
and F respectively.

Shade or color in the triangle with vertices A, B, and the center. Then shade in the
triangle whose vertices are at D, F, and the center, and the triangle whose vertices
are at C, E, and the center.
Reciprocal Identities: The
Pythagorean Identities: For
Product Identities: Along the
two trigonometric functions
each shaded triangle, the upper-
outside edge of the hexagon, any
on any diagonal are
left function squared plus the
trigonometric function equals the
reciprocals of each other.
upper-right function squared
product of the functions on the
equals the bottom function
adjacent vertices.
squared.
Trigonometry
Unit 5 Packet
Verifying Identities1
Hints for verifying identities:
1. If one side of an equation looks more complex, start there.
EXAMPLE: tan 2 x(1  cot 2 x) 
1
1  sin 2 x
2. Try changing everything into sin/cos.
EXAMPLE: cot x  1  csc x  cos x  sin x 
1
WARNING: Do NOT handle the identities to be established as equations. You cannot add the same expression to both
sides, etc.
Trigonometry
Unit 5 Packet
3. Factor.
Example: sec4 x  sec2 x  tan 4 x  tan 2 x
4. Create a common denominator.
EXAMPLE:
1
1

 2sec2 x
1  sin x 1  sin x
Trigonometry
Unit 5 Packet
Sum and Difference Identities
tan( A  B) 
tan A  tan B
1  tan A tan B
tan( A  B) 
tan A  tan B
1  tan A tan B
EXAMPLE 1: Find the exact value of each expression.
a)
b)
c)
d)
( )
e)
f)
EXAMPLE 2: Suppose that
and both and
are in quadrant II. Find
Trigonometry
Unit 5 Packet
Double-Angle Identities
EXAMPLE 1: Given sin   
2
(QIII) find:
3
cos 2
sin 2
EXAMPLE 2: Given that sin  
EXAMPLE 3: Simplify
EXAMPLE 4: Simplify
2 tan
2
and cos   0 , find tan 2 .
5

3 .
1  tan 2

3
.
tan 2
Trigonometry
Unit 5 Packet
Half-Angle Identities
√
√
√
EXAMPLE 1: Find the exact value of:
a)
EXAMPLE 2: Given
b)
, with
, find:
a)
b)
EXAMPLE 3: Simplify: 
1  cos12 x
2
EXAMPLE 4: Evaluate tan
9
.
8
EXAMPLE 5: Let csc   5 (QIII). Find tan

2
.
Trigonometry
Unit 5 Packet
PRACTICE:
Match the term in column I with the term in column II that completes the identity.
Answer
I
1.
cos x
sin x
A.
2.
B.
3.
C.
4.
D.
1
sec 2 x
5.
E.
6. –
sin 2 x
F.
cos 2 x
sin x
G.
cos x
7.
8.
sec x
csc x
H.
9.
I.
10.
J.
11. Suppose a student writes, “
12. Another student claims, “Since
both sides to get
II
” What is wrong with this statement?
, you can take the square root of
.” TRUE or FALSE? Explain.
Trigonometry
Unit 5 Packet
Find the EXACT value:
13. cos165
15. sin105
14. cos
16.
17.
tan 20  tan 25
1  tan 20 tan 25
18. tan
19.
2 tan15
.
1  tan 2 15
20.
21. 2cos2 15  1
23. cos 22.5
7
12
7
12
tan 34
.
2 1  tan 2 34

22. cos

7
8
24. tan 22.5
Trigonometry
25. sin
Unit 5 Packet
3
8
26.
27. Let sin   
1  cos 59.74
sin 59.74
4
1
(QIV) and cos  
(QI). Find cos    
5
5
28. Suppose that A and B are angles in standard position, with
, and
. Find each of the following:
a)
b)
c) the quadrant of
SIMPLIFY the expressions:
29.
31. 
30.
1  cos 5
sin 5
1  cos8
1  cos8
32. Let sin   
1
(QIV). Find cos 2 .
3
33. Let cos  
3
(QI). Find sin 2 .
5
Trigonometry
Unit 5 Packet
34. Given that cos  
find tan 2 .
3
and sin   0 ,
5
36. Let sec  2 (QIV). Find:
sin

cos
2
37. Let sin   
sin
5
and
7
cos   0 , find cos 2 .
35. Given that sin   

tan
2

2
1
(QIV). Find:
3

cos
2

2
38. Is this problem solved correctly? If not, correct the error(s):
Problem: Solve
.
tan

2
Trigonometry
Unit 5 Packet
EXPANSION 1: Using the information in the picture below, write an expression to
represent tan

4
.
b
50
50
EXPANSION 2: Prove that cos      cos      2cos  cos  .
EXPANSION 3: Given that cos

12

1
4


6  2 , evaluate cos

24
.
EXPANSION 4: Use the fact that tan 3x  tan  2 x  x  to establish an identity for tan 3x .
EXPANSION 5: Show that the area A of an isosceles triangle whose equal sides are of
length s and the angle between them is θ is: A 
1 2
s sin  .
2
s
θ
s

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