HAT
NOTES
Chapter 7
(skip 7.5)
(Spring 2010)
Honors Algebra Trigonometry Chapter 7
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Honors Algebra Trigonometry Chapter 7
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Revisiting Verifying Identities
Reminder: tanx =
secx =
cscx =
Pythagorean Identities: Using the unit circle:
The basic/most useful one:
Goal: Make one side of the equation look like the other. Change one side ONLY!!!!
As a hint, work with the more complicated side.
1)
sin   tan 
=
sec  - cos 
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2) tanx + cotx
3)
=
secx  cscx
1  csc3
- cot3
sec3
=
cos3
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Partner Practice:
Verify the following identies.
4
4) cos 2
5)
+ sin 2 2
1  sec 4 x
sin 4 x  tan 4 x
= cos 2 2 + sin 4 2
=
Honors Algebra Trigonometry Chapter 7
csc4x
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Let’s do these together, as a group.
6)
7)
cos 
sec   1
csc 
cos
sec -1
+
+
1
=
Honors Algebra Trigonometry Chapter 7
=
2cot 2
cot 2
csc - 1
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8)
cos3 x  sin 3 x
 1  sin x cos x
cos x  sin x
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Honors Algebra Trigonometry Chapter 7
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Warm-Up: Solving Equations
#1-3: Graph each equation from 0 to 2  .
1) y = sinx
2) y = sin(2x)
1
3) y = sin( x )
2
Go back to the graphs in #1-3 and place a dot where the y value(s) is (are)
Honors Algebra Trigonometry Chapter 7
1
.
2
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Solving Trig. Equations
Find all values for each variable {We will need to use ‘n’}.
1
1
1a) sinA =
2a) sin(2B) =
2
2
Unit Circle:
Find all values for each variable from [0, 2  ].
1
1b) sinA =
2
Honors Algebra Trigonometry Chapter 7
2b) sin(2B) =
1
2
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Find all values for each variable {We will need to use ‘n’}.
3a) sin(
D
1
)=
2
2
4a) sin(
D
1
)= 
2
2
Find all values for each variable from [0, 2  ].
D
1
D
1
3b) sin(
)=
4b) sin(
)= 
2
2
2
2
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Find all values for each variable {We will need to use ‘n’}.
3
2
5a) sinC =
6a) sin E = 
2
2
Find all values for each variable from [0, 2  ].
3
5b) sinC =
2
Honors Algebra Trigonometry Chapter 7
6b) sin E = 
2
2
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II. #7-8: Review Factoring: Solve for the unknown variable.
7) x2 - 1 = 0
8) 2y2 + 3y - 2 = 0
III. Let’s tie the factoring into what we are currently doing….
Solve for all values of y.
9) sin2y - 1 = 0
10) 4sin2y - 3 = 0
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#11-14: Solve for the unknown variables over the interval of : [ 0 , 2  ).
11) sinw cosw = 0
12) 2sin2x + 3sinx - 2 = 0
13)
2sinxcosx = cosx
Honors Algebra Trigonometry Chapter 7
14) secxcscx=2cscx
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III. Find the general equation(S) that represents all possible solutions. n = integers
15) cotx = cosx
Honors Algebra Trigonometry Chapter 7
16) 2sin2x - cosx - 1 = 0
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Solve for the unknown variables over the interval of : [ 0 , 2  ).

17) sin(3x - )= 1
4
18) tan2x = 1
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19) sin(2x-  ) = 0

2
20) cos4(x+ ) = 1
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21) cosx + 1 = sinx
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7.3: Cofunctions:
Warm-up: #1-2:
1) Complete the sides of the triangles below.
1
45o
2)
1
1
30o
60o
Introduction to cofunctions…. Find the acute angles: A – D.
3
3
a) sinA=
and
= cos (B)
Find A and B
2
2
b) sinC= 0.342 = cos(D)
Find C and D
Notice: sinA = cos (
)
sin C = cos (
)
cosB = sin(
)
cos D = sin (
)
sine and cosines are called cofunctions of each other.
c) What do you think the cofunction of cscA is? ___________
cscA = _________________________
Let’s find out about tan and cot….Do the same rules apply? Let’s check with an
example

 
d) Check: Is tan = cot( - )?
6
2 6
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Cofunctions Formulas:
cos (

- u) = sinu
2
sin(

- u) = cosu
2
tan (

- u) = cotu
2
cot(

- u) = tanu
2

- u) = cscu
2
csc(

- u) = secu
2
sec(
Express as a cofunction of a complementary angle. All given angles are in radians….
3) sin(

)
7
5) tan(2)
Honors Algebra Trigonometry Chapter 7
4) cos(
1
)
7
6) sec(1.2)
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Sum and Difference of Angles
Warm Up: What are our ‘magic angles?’ (the angles we know the exact values for?)
1) In degrees:
2) In radians:
Let’s find the basic values using the unit circle:



3)  
4)
6
4
5)  
sin  
sin  
sin  
cos  
cos  
cos  

3
So far, we have found exact values of angles with reference angles of 30, 45 or 60. What
happens if we don’t have one of these angles?????
6) Find the exact value of sin(105o).
Ok- so this is a problem…. Since this isn’t one of our ‘magic’ numbers, we can’t seem to
do this yet. But, wait!!! There is probably some formula somewhere that could help,
let’s look on the next page and see what we can learn.
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Sum/difference formulas:
sin (    ) =
cos (    ) =
7) Let’s go back and do sin(105o) using our new formulas….
8) Let’s look at the cofunction formulas as see how it works with the sum and difference
formulas:


cos ( - u) = sinu
sin( - u) = cosu
2
2
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Here are a few more examples.
9) sin(15o)
10) sin(

)
12
11) cos ( 195  )
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12)
cos( -75o)
13) Find the value of: cos20 o  cos70 o – sin20 o  sin70 o
14) Find the value of sin(

6

6
)cos(
) + cos( )sin(
)
7
7
7
7
15) How would problem #14 change if it had been: cos(
Honors Algebra Trigonometry Chapter 7

6

6
)sin(
) + sin( )cos(
) ?
7
7
7
7
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16) Verify that: sin( 270o + x ) = -cosx
17) If  and  are acute angles such that cos  =
Honors Algebra Trigonometry Chapter 7
7
17
and csc  =
, find sin(  +  )
15
25
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Sum and Difference with tangents…
tan(u+v) =
tan(u-v) =
__tanu + tanv___
1 – tanu tanv
__tanu – tanv
1+ tanu tanv
Practice:
18)
tan


- tan
4
6
19)
tan

12
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Double Angle/Half Angle Formulas
1) Warm-up: Use the sum and difference formulas to simplify the following:
a) sin ( x + x )
b) cos ( x + x )
c) tan(x + x )
Formulas/Identities:
cos2  = cos 2  - sin 2 
sin2  = 2sin  cos 
cos2  = 1 - 2sin 2 
cos2  = 2cos 2  - 1
tan2  =
2 tan 
sin 2
=
2
1  tan 
cos 2
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Examples: The first example will use numbers that we are familiar with, to verify that the
formulas work.
1
2). Given:
sinx =
, x is in the 2nd quadrant.
2
cosx =
{ we had 2 ways to do this, the easiest is to draw the
triangle ( remember quadrant) and solve for the 3rd side }
tanx =
a) cos2x
b) sin2x
c) tan2x
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3) Given: cos  =
4
.  is in the 4th quadrant. {Draw the angle}
5
a) sin  =
b) tan 
c) sin2  =
d) cos2  =
e) tan2 
Let’s discuss restrictions. ( Which quadrant the new angle will lie in )
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4) cos  = a)
1
.  is in the 3rd quadrant. Find cos2  and sin2  and tan2  .
5
sin 
b) cos2 
c)
sin2  .
d) tan2 
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Half Angle Formula/Identity
cos
x
= 
2
1 + cosx
2
sin
x
= 
2
1 - cosx
2
tan
x
2
tan
x
2
x
1 - cosx
2
=

1+cosx cos x
2
or
1  cos x
sin x

=
sin x
1  cos x
sin
Examples:
5) Given: sinx =
1
, x is in the 1st quadrant.
2
Restrictions on x ?
Restrictions on
x
?
2
a) cosx =
b) cos
x
=
2
d) tan
x
=
2
Honors Algebra Trigonometry Chapter 7
c) sin
x
=
2
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6) Given: cosx =
4
,
5
x is in the 4th quadrant.
Restrictions on x ?
a) cos
x
=
2
b) sin
x
=
2
c) tan
Restriction on
x
?
2
x
=
2
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7) Verify the identity: cos4x – sin4x =
Honors Algebra Trigonometry Chapter 7
cos2x
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8) cos6x
=
Honors Algebra Trigonometry Chapter 7
32cos6x – 48cos4x +18cos2x – 1
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Inverses
Warm-Up: Complete #1 – 3….
1) Reminder: Find the inverse of the following functions.( x for y and y for x)
a. f(x) = 3x – 6
b. f(x) = x2 + 6
(x<0)
Facts about inverses of functions as learned in section 3.8: p. 530
They must be 1-1 functions {if a  b, in the domain of f, then f(a)  f(b)}
Horizontal line test….
If x = f(y) , then y = f-1(x)
Domain of the inverse, f-1= range of the function ,f.
Range of the inverse, f-1 = domain of the function,f.
f(f-1(x)) = x for every x in the domain of f-1
The graphs of f and its inverse are reflections of each other through y = x.
2) Use your calculator to find the unknown angle measure, to the nearest hundredth of a
degree, for each equation.
a. sinA = 0.348
b. cosB= 0.998
3) Using your knowledge of functions, what is the inverse of each of the following trig.
functions?
a. sinx = y
b. cosx=y
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4) Draw a rough sketch of a graph of a sine curve and a cosine curve below.
Are these 1-1 functions?????
How can we make them 1-1 functions?
5) What do we know about the domain and range the inverse of each function from #4?
y = sin-1x
y = cos-1x
sinx:
cosx:
sin-1x
cos-1x
Example: Solve for all values of y in the equation: y= sin-1(1/2)
The term ‘arcsine’ is equivalent to the term inverse. In other words…
sin-1x = arcsin(x)
or
cos-1x = arccosx
Angle = sin-1(of an number between or = to 1 and -1)
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Visual Picture of the inverses:
Sine
cosine
Write an equivalent statement for each function. If the statement is in inverse form, find one that
does not and visa versa.
6) y = sin-1a
7) y = arcsinb
8)
3
= cosc
2
9) y = arccos(0)
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Example Problems:
Find the exact value of the expression whenever it is defined.
2
1
10) sin-1(  )
11) cos-1( 
)
2
2
12) arcsin(0)
14)
13) arccos(
2
sin(sin-1 )
3
14b) cos(sin-1
2
)
3
Honors Algebra Trigonometry Chapter 7
2
)
3
15) cos-1(cos(
15b)

))
6
cos-1(cos(
11
))
6
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16) sin(cos-1(
3
))
2
17) sin-1(cos
5
)
6
18) cos-1(cos
7
)
6
19) sin-1(sin
7
)
6
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The inverse of tangents, or arctangents…..
20) Sketch of graph: y = tanx
21) Domain for 1-1 function: (find a section that passes the horizontal line test and is
continuous.
Domain of tanx:
Range of tanx:
Domain of tan-1x:
Range of tan-1x:
Examples:
22) tan-1(-1)
24) arctan(tan(
23) tan[tan-1(-9)]

))
4
26) tan(cos-1(0))
Honors Algebra Trigonometry Chapter 7
25)
tan-1(tan
7
)
6
1
27) csc(cos-1( ))
5
Page 162
28) sin(2tan-1
5
)
12
29) tan[2arcsin(
8
)]
17
1
4

30) cos sin 1 12
13  tan ( 3 )
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Honors Algebra Trigonometry Chapter 7
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Review Chapter 7
HAT
1) Simplify:
(sec u  tan u )(csc u 1)  cot u
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2) Verify the identity: tan   cot   2 csc2
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3) Find all solutions of the equation:

2

cos  4 x   
4 2


2

3.5) Solve : cos  4 x   
4 2

from [0,2 )
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4) Find the exact value of tan
7
.
12
5) Find the exact values of sin2 for the given value of  :
4
sin    ; 2700    3600
5
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6) Find the exact values of cos2 for the given value of  :
4
sin    ; 2700    3600
5
7) Find the exact values of tan2 for the given value of  :
4
sin    ; 2700    3600
5
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 
8) Find the exact values of sin   for the given conditions.
2
csc   
5
;  900    00
3
 
9) Find the exact values of tan   for the given conditions.
2
5
csc    ;  900    00
3
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10) Find the exact value of the expression whenever it is defined:

arcsin
2
11)
Find the exact value of the expression whenever it is defined:
cos1  cos 43 
12)
Find the exact value of the expression whenever it is defined:

3
arctan  

 3 
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13) Find the exact value of the expression whenever it is defined:
5
sin sin 1 13
 cos1( 53 )
14) Verify the reduction formula using the sum/difference formulas:
3 

sin   
  cos 
2 

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15)Express cos3 in terms of cos .
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16) Find the exact value of: cos sin 1( 45 )  tan 1( 43 )
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