Download PRECALCULUS PREAP CALENDAR UNIT 7

PRECALCULUS PREAP CALENDAR
UNIT 7
January 5 – DECEMBER 18
Date
Assignment
Tuesday
1-5
Wednesday
1-6
Thursday
1-7
Friday
1-8
Monday
1-11
Tuesday
1-12
Graphing Inverse Trig Functions and Day 1 of Inverse
Values
Wednesday
1-13
Thursday
1-14
Friday
1-15
Reference
Inverse Values Day 2
Analytical Trigonometry
Simplifying Trig Expressions
Simplifying Trig Expressions
More Simplifying Trig Expressions
Intro Proving Trig Identities
Proving Trig Identities
More Proving Trig Identities
Review
Test over Inverse Functions and Trig Identities
1
INVERSE SINE PARENT GRAPH
Graph y = sin −1 ( x ) (the inverse function of sin x )
Alternate notation is y = arcsin ( x )
For y = sin −1 ( x ) , answer the following questions.
1) What is the domain?
2) What is the range?
3) So what angles are allowed with this function?
Examples:
Graph each function using your knowledge of transformations. State the domain and range.
1) y sin −1 ( x − 1)
=
2) y = sin −1 ( 2 x )
Domain:
Range:
Domain:
Range:
y
y
2π
2π
3π
2
3π
2
π
π
π
2
π
2
x
−4
−3
−2
−1
1
2
3
4
x
−4
−3
−2
−1
1
−π
2
−π
2
−π
−π
−3π
2
−3π
2
−2π
−2π
2
3
4
2
=
3) y arcsin x −
π
=
4) y
4
Domain:
Range:
1 −1
sin ( − x )
2
Domain:
Range:
y
y
2π
2π
3π
2
3π
2
π
π
π
2
π
2
x
−4
−3
−2
1
−1
2
3
4
x
−4
−3
−2
−1
1
−π
2
−π
2
−π
−π
−3π
2
−3π
2
−2π
−2π
2
3
4
------------------------------------------------------------------------------------------------------------------------------INVERSE COSINE PARENT GRAPH
Graph y = cos −1 ( x ) (the inverse function of cos x )
Alternate notation is y = arccos ( x )
For y = cos −1 ( x ) , answer the following questions.
1) What is the domain?
2) What is the range?
3) So what angles are allowed with this function?
3
Examples:
Graph each function using your knowledge of transformations. State the domain and range.
=
1) y cos −1 x +
π
2) y = − cos −1 ( x )
2
Domain:
Range:
Domain:
Range:
y
y
2π
2π
3π
2
3π
2
π
π
π
2
π
2
x
−4
−3
−2
−1
1
2
4
3
−4
−3
−2
−1
1
−π
2
−π
2
−π
−π
−3π
2
−3π
2
−2π
−2π
y cos −1 ( − x ) +
4) =
3) y 2arccos ( x + 1)
=
Domain:
x
Range:
2
3
4
π
4
Domain:
Range:
y
y
2π
2π
3π
2
3π
2
π
π
π
2
π
2
x
−4
−3
−2
−1
1
2
3
4
x
−4
−3
−2
−1
1
−π
2
−π
2
−π
−π
−3π
2
−3π
2
−2π
−2π
2
3
4
4
INVERSE TANGENT PARENT GRAPH
Graph y = tan −1 ( x ) (the inverse function of tan x )
Alternate notation is y = arctan ( x )
For y = tan −1 ( x ) , answer the following questions.
1) What is the domain?
2) What is the range?
3) So what angles are allowed with this function?
Examples:
Graph each function using your knowledge of transformations. State the domain and range.
=
1) y tan −1 x −
3π
2
2) y = 2 tan −1 ( x )
Domain:
Range:
Domain:
Range: _____
y
y
2π
2π
3π
2
3π
2
π
π
π
2
π
2
x
−4
−3
−2
−1
1
2
3
4
x
−4
−3
−2
−1
1
−π
2
−π
2
−π
−π
−3π
2
−3π
2
−2π
−2π
2
3
4
5
=
3) y 2arctan x −
π
4) y =
− tan −1 ( x + 1)
4
Domain:
Range:
Domain:
Range:
y
y
2π
2π
3π
2
3π
2
π
π
π
2
π
2
x
−4
−3
−2
−1
1
2
3
x
4
−4
−3
−2
−1
1
−π
2
−π
2
−π
−π
−3π
2
−3π
2
−2π
−2π
2
3
4
GRAPHING INVERSE TRIG FUNCTIONS
Find the domain, range, and sketch a complete graph of each function. Inverse functions are denoted by
y = sin −1 x or by y = A rcsin x .
2) y = cos–1(x) -
π
1)
y = sin –1(3x)
5
y = 3 arccos(2x-4) 6) y = tan –1(x-1) + π 7) y = – arc sin x +
8)
y = 3 cos–1 (x-2)
9) y = -
2
1
tan–1 (x-1)
4
3) y = arc sin (x+1) 4)
π
2
x
3
y = 2 sin –1 ( )
10) y = tan–1 x +1
6
NOTES
REVIEW FINDING INVERSE TRIG VALUES AND ANGLES
FIRST GROUP
y = sin −1 x
 π π
 − 2, 2 


y = csc −1 x
 π π
 − 2, 2  , y ≠ 0


y = tan −1 x
 π π
 − 2, 2 


Sample Problems
Triangle problems


Draw a reference triangle to determine the value of each function.
3

Angle Problems




2) cot  sin −1  −
1) sin  arctan 
5
4
sin −1 (
1 

10  
3
)
2
3) cos ( arctan x )
5 arc tan(-1)
6 arcsin(tan
3π
)
4
SECOND GROUP
y = cos −1 x
[0, π ]
y = cot −1 x
(0, π )
y = sec −1 x
[0, π ] y ≠
π
2
Sample Problems
Triangle Problems Draw a reference triangle to determine the value of each function.


3
4) sin  arccos 
5

5)


5 
sin  cos −1  −
 

 5 

6) tan ( arcsin 2x )
7
Angle Problems
4
cos −1 (
3
)
2
5 arc cot(-1)
6 arcsin(cos
3π
)
4
Summary of quadrant locations for inverse functions.
Examples:
Find the exact values without using a calculator.

3
cos −1  −
 = __________
 2 
1.
sin −1 (1) = ________
3.
1
sin −1   ________
2
5.
tan −1 − 3 = ________
6.
 2 
csc −1 
 = __________
 3
7.
Arc csc 2 = ____________
8.
sec−1
9.
sec −1 − 2 = _____________
10.
 1 
A rc cot  −
 = _______________
3

(
(
)
)
2.
4.
1
A rccos   = __________
2
( 2 ) = ____________
8
INVERSE VALUES AND ANGLES
Find the exact value of the expression using radicals or radians if necessary.
5

1. sec  sin −1 
7


 3 
2. cot  sec −1  −  
 2 

3. csc ( tan −1 ( −5 ) )
  π 
4. sec −1  sin  −  
  6 
  4π  
5. sin −1  sin 

  3 
Composition of Trig Values


5 
6. sec  csc −1  −
 

2



Find the exact values of each expression using radicals or radians if necessary.



 4 
 4 
 5 
1. tan  cos −1   
2. cos  arctan   
3. sin  tan −1  −  
 5 
 3 
 12  



  π 
6. arccos  sin   
  6 

 7π  
9. sin −1  cos 

 6 


 4 
4. sec  arcsin  −  
 7 

  4π  
7. cos −1  sin 

  3 
5. sin −1 ( cos ( 0 ) )
10. tan −1 ( cos (π ) )

 4π
11. tan −1  tan  −
 3


 4 
13. sin  sec −1  −  
 3 

  π 
14. arcsec  sec  −  
  3 
16. tan arcsec − 2
17. sin ( sec −1 ( −4 ) )
18. sec ( csc −1 ( −3) )
20. arcsin ( cos (π ) )

 7 
21. tan  sin −1  −  
 5 

(
19.
(
))
csc ( cot −1 ( 2 ) )

 5 
8. cot  csc −1  −  
 3 






3 
12. cos  arcsin  −
 

5



  3π  
15. sin −1  cot   
  4 
9
INVERSE TRIG PROBLEMS
Evaluate each of the following expressions. You are finding the value of each trig expression.
15 

17 
1) cos  sin −1

6) cot  tan −1 
10

9) csc  tan −1

−15 

8 
−1 

3 
13) csc  tan −1


5 

13 
1
6
5) sin  csc −1 
5

2) sin  cos −1


10) tan  sec −1
−13 

12 


3) cos  sin −1

−3 

5 
12
7) sec  cot −1 
5 


11) tan  sec −1

− 13 

3 

3
4) sin  cos −1

3 

5
8) tan  csc −1 
3


12) cos  cot −1

−3 

2 
− 10 
14) cot  sin −1

10 

Write and algebraic expression for the given expression.
15) tan ( arctan x )
16) sin ( arccos x )
17) cos ( arcsin 2x )
18) sec ( arctan 3x )
x
19) tan  arc sec 
20) cot ( arc cot x )
3

Part II
Use a reference triangle to find the value.
1
1) sin  arccos 
2


1
4) cos  arcsin 
4


−3
2) sin  arccos 
5 

3) tan  arcsin

2 

5) sec ( arc csc a )
x
6) cot  arc csc 

2
3

Find the angle.
7) arcsin( −1)
−1
10) arccos  
 2 
3π
13) arccos  cot 
2 

− 2
8) arcsin 

 2 
9) arccos( −1)
π
11) arccos  cos 
3

3π 

14) arcsin  tan 
4 


4π
12) arcsin  sin 
3 

15) arcsin  sin

−2π 

3 
10
Fundamental Trigonometric Identities
Reciprocal Identities:
Quotient Identities:
Pythagorean Identities:
Negative Angles
Trig Identities Worksheet
Day 1 Simplifying Expressions
Simplify the given expression to a single trig term or a single number. Circle your answer.
1)
 cos2 θ + sin2 θ

cos θ

3)
 1 


  sec θ 
2)
 sin θ 

 ( cot ( −θ ) )
 cos θ 
(1 + cot θ )(1 − cos θ )
4)
( cos β )( sec
5)
tan φ  csc φ
6)
sin θ ( csc θ − sin θ )
7)
cot θ  sec θ
8)
sec2 β (1 − sin2 β )
9)
cot φ  sin φ
10)
1 + tan2 θ
1 + cot2 θ
11)
sec θ − ( tan θ  sin θ )
12)
 cos2 θ

 sin θ
2
2
2
2
β − 1)

 + sin θ

11
sin θ + tan θ
1 + sec θ
13)
(1 − sin θ )(1 + tan θ )
16)
( sin θ + cos θ )2 + ( sin θ − cos θ )2
18)
sin2 θ
+ cos θ
cos θ
21)
sec2 θ
sec2 θ − 1
22)
tan2 θ + 1
tan θ  csc2 θ
24)
2 + cot2 θ
−1
csc2 θ
25)
cot2 θ − ( cos 4 θ  csc2 θ )
2
2
14)
19)
17)
15)
sec θ tan θ
−
cos θ cot θ
cos β  csc β
1
1
+
2
sec θ csc2 θ
20)
23)
sec θ  sin θ
tan θ + cot θ
cot θ
+ tan θ
csc θ + 1
PROVING TRIG IDENTITIES
1)
sin x + cos x cot x = csc x
2)
cos x csc x = cot x
3)
2 cos2 x - sin 2 x + 1 = 3 cos 2 x
4)
sin x(sec x - csc x) = tan x - 1
5)
1
1
+
=
2 sec2 x
1 + sin x 1 − sin x
6)
sin2 x
+ cos x =
1
1 + cos x
7)
sin x cot x + cos x
= 2 cot x
sin x
8)
sin x cos x
1
=
2
1 − 2 sin x cot x − tan x
12
9)
10)
1 + tan2 x
= csc2 x
tan2 x
1 - tan x
1 + tan x
= cot x - 1
cot x + 1
11)
1 − sin2 x
= cot2 x
1 − cos2 x
12)
sec x
= sin x
tan x + cot x
13)
sin2 x
= sec2 x − 1
cos2 x
14)
sin x
- sin x cos x
1 - cos x
1 + cos x
= csc x + csc x cos2 x
VERIFY THE FOLLOWING IDENTITIES.
1)
csc2 x(1 − cos2 x) =
1
2) (sin x + cos x)2 − (sin x − cos x)2 =
4 sin x cos x
3)
sin x(csc x + sin x sec2 x) =
sec2 x
4)
cot2 x +=
5 csc2 x + 4
5)
cos 4 x − cos2 x = sin 4 x − sin2 x
6)
sinxtanx + cosx = secx
7)
− cos2 x
sin x − csc x =
sin x
8)
1
1
+
= cos x − sec x
sec x cos x
9)
sinx + cosxcotx = cscx
10)
(sin x − cos x)2
= sec x − 2sin x
cos x
11)
sec x
= sin x
tan x + cot x
12)
cos x
1 + sin x
=
1 − sin x
cos x
13
PRECALCULUS PREAP REVIEW
Part 1 Find the domain and range for the following inverse functions.
1)
=
f ( x ) 2arcsin x − 2π
1
f ( x ) = arccos3 x
2
2)
3)
=
f ( x ) tan −1 ( 3 x ) +
π
2
Part 2 Sketch the graph
4)
y = 3sin −1 2 x
5)
y = 2arctan x
1
=
y arccos x + π
6)
2
Part 3 Simplify the following expressions. Give exact values.
2
2
7)
arccos
11)

3
tan  cos −1

2 

8)
12)
tan −1 ( −1)

3
sin  cot −1

4 

9)

 π 
cos −1  cos  −  
 4 

10)
 π
sin −1  sin 
5

13)
tan ( sec −1 3x )
14)
x

csc  arccos 
3

Part 4 Simplify the expression
sec x
tan x + cot x
15)
1
1
+
2
sec x csc 2 x
16)
19)
1 + tan x
1 + cot x
20) sin x sec x
17) (1 − tan 2 x )(1 − sin 2 x )
21) sec x − sin 2 x sec x
18) sec x − sin x tan x
22)
cos 2 x
1 − sin x
Part 5 Prove the identities
23) sin x ( sec x − csc x ) =tan x − 1
24)
1 − cos x
sin x
+
=
2csc x
sin x
1 − cos x
25)
tan x sin x
tan x − sin x
=
tan x + sin x
tan x sin x
14