Download Unit 7 - Inverse Trig Functions PAP Packet

Pre-AP/GT Pre-Calculus Assignment Sheet
Unit 7 – Inverse Trig Functions
January 19th – 25th, 2017
Thursday
1/19
Graphing Inverse Functions
Pages 6-7
Evaluating Trig functions
Page 10
Monday
1/23
Evaluating Trig functions
Quiz on Inverse Functions
Page 12
Tuesday
1/24
Solving Trig. Equations with Calculator and by hand
Page 14
Friday
1/20
Review
Wednesday
1/25
Study!
Unit Test
1
Notes on Graphing Inverse Trig Functions
Let’s start by graphing Sine, Cosine, and Tangent below:
Sine
Cosine
Tangent
Are these functions considered One-To-One? Why or why not?
If a function is not One-To-One, their domains must be restricted (this will allow their inverses to be a
function).
How to find the inverse of a Trig Function:
Function
Sketch
Inverse
Domain
Range
Notice: When we found the Domain and Ranges for the inverse functions, we switched the functions Domain
and Range. This comes from Algebra II when you first learned to find the inverse (switch x and y). Next, let’s
sketch the graphs of the inverses.
2
We can use the domain and ranges to sketch the graphs of the inverse functions
y  arcsin( x) or y  sin 1 ( x)
y  arccos( x) or y  cos 1 ( x)
   
,
Domain:  1, 1  Range: 
 2 2 
Domain:  1, 1  Range: 0, 

y  arctan( x) or y  tan 1 ( x)
   
, 
Domain:  ,   Range: 
 2 2
Having restricted the interval on which we graph so that each inverse is a function results in only one answer
for each problem. The range of sine and tangent is in Quadrants I and IV, while the range of cosine is
Quadrants I and II. Label this information on a coordinate plane below.
3
Graph the following Inverse Trig Functions: State the Domain and Range.
1. y  sin 1 (2 x)
2. y  arccos( x) 
4. y  cos 1 (2 x  2)  
x
5. y  arctan( )
2

2
3. y  arcsin( x  1)
6. y  sin 1 ( x  2) 

2
4
Assignment on Graphing Inverse Trig functions
Show all work.
Graphing Inverse Trig Equations: Sketch a graph of each of the following and state the domain and range
1. y = sin-1(3x)
2. y =
3. y = arcsin(x + 1)
4. y = 2sin-1(x)
5. y = arccos(2x – 4)
6. y = tan-1(x) + π
5
Graphing Inverse Trig Equations: Sketch a graph of each of the following and state the domain and range
x
7. y  arcsin( )
2
9. Let sin x 
8. y 

2
 cos 1 ( x  2)
5
. Find the exact values of all six trig functions. (Hint…Draw a Right Triangle)
7
10. Solve the following for x:

a. sin 1 x 
4
b. arccos( x) 
5
6
c. tan 1 x  0
6
Notes on Inverse Functions (Day 1)
Review of Domain Restrictions and Quadrants:
arccos x
arcsin x
arctan x
Inverse Trig Functions - Draw a reference triangle and evaluate each of the following expressions. Remember to be
careful of which quadrant.
1
1. sin  arccos 


2
5
3. tan  arccos

6 



3
2. sin  arccos 
5


4. cos  arc csc

13 

5 
7
Find the exact values without using a calculator.
1. sin 1 1 =
3. sin 1 1.5  =
2. Arc csc2 =
1
2
4. A rccos   =

5. A rc cot  

1 
=
3
Find the exact value or angle in terms of  . Remember to look at the outside function to determine if the
answer will be an angle or value.


1
2
1. sin  A rccos 


4. sin 1  cos
5 

4 
  

 4 

2. Arc sec  sec  




 3 
 4  


5. sin  cos 1 
3. tan 1  sin  5  

 1 

 2 
6. tan  sin 1 

8
Assignment on Inverse Functions (day 1)
Show all work.
Find the exact values without using a calculator.

 2 

 3
2. sec1  2
1. csc 1 




3. cos 1  
3

2 
Inverse Trig Functions - Draw a reference triangle and evaluate each of the following expressions. Remember to be
careful of which quadrant.
15
5

1. cos  sin1

17


1




3
9. tan  arcsin 
5

3


13 

3. sin  cos1

3 

12 
6. sec  cot 1

5 

7. tan  sec1

12


5. cot  tan1

10


2. sin  cos1

13

6
4. sin  csc1 
5

15 

8. csc  tan1

8 

1
10. cos  arcsin 
4


Find the exact value or angle in terms of  . Remember to look at the outside function to determine if the
answer will be an angle or value.


11. A rcsin  sin

7 

6 
12. sin 1  sin
 2 
 7 
16. sin tan 1(1)
15. cos  A rcsin    






 3  

 2 
13. cos 1  cot  

4



 5 

 13  
17. cos  A rcsin  




 2 
 9 
14. csc cot 1  1
18. csc  cos 1   

9
Notes on Inverse Trig Functions (Day 2)
Use an Inverse Trigonometric to write  as a function of x.
1.
2.
Properties of Inverse Trig Functions

sin(arcsin x)  x
and
sin(arcsin y )  y
iff
 1  x  1 and 
cos(arccos x)  x
and
cos(arccos y )  y
iff
 1  x  1 and 0  x  
tan(arctan x)  x
and
tan(arctan y )  y
iff
 1  x  1 and 
2

2
x
x

2

2
Evaluate the following:
1. sin(arcsin 0.6)
2. tan(arctan 35)
Write an Algebraic Expression that is equivalent to the expression.
1. sin(arctan x)
x
3
3. cot(arccos )
2. sec(arctan 3x)
4. sec(arcsin ( x  1))
10
Notes on Solving Trig Equations with Calculators
Determine the values of  , where 0    360 , to the nearest hundredth of a degree.
Before you begin: Make sure you are in DEGREE MODE!!!!!! On your calculator.
Determine the reference angle using your calculator. Where could the angle lie? Quadrant I, II, II, IV
Find both angle values of  .
1. sin  = .7183
2. tan  = 1.6198
3. cos  = – .6691
4. sec  = – 4.8097 (2nd cos 1/– 4.8097 )
5. cot  = – .1228
(2nd tan 1/– .1228)
Determine the values of  , where 0    2 , to the nearest hundredth of a radian.
Before you begin: Make sure you are in RADIAN MODE!!!!!! On your calculator.
6. sin  = – .8183
7. tan  = 2.4567
8. csc  = – 1.1859
Fun Ones: Solve the following:
9. 3 sec  12  sec  21
10. 3 cos2   2 cos  1  0
11
Assignment on Solving Trig Equations with Calculators
Determine the values of  , where 0    360 , to the nearest hundredth of a degree.
1. sin  = 0.4067
2. cos  = – 0.5023
3. tan  = 2.9988
4. sec  = 1.1111
5. cot  = – 1.2222
6. csc  = 2.5012
Determine the values of  , where 0    2 , to the nearest hundredth of a radian. (Radian Mode)
7. sin  = 0.8143
8. cos  = 0.7838
9. tan  = –.2677
10. csc  = 1.0204
11. cot  = 0.5890
12. sec  = – 1.5861
Solve each of the following on the interval from [0, 2π)
13. 11csc x + 15 = 9csc x + 19
14. 2cos2x – 1 = 0
15. 2 cos 2 x  3  0
16. 2 sin2 x  5 sin x  3  0
12
Notes on Solving Trig Equations by Hand
Solve the following equations on the interval 0    2 . Give the exact answer in terms of  .
1. 2sin   1  0
4.
2 csc  2  0
7. 2 sin 2   sin   0
3 cot   1  0
2. 4 tan x  4  0
3.
5. tan   3  2 tan 
6. sec 2   2  0
8. 2 sin 2   sin   1  0
13
Assignment on Solving Trig Equations by Hand
Solve the following equations on the interval 0    2 . Give the exact answer in terms of  .
1. tan   3  0
3 csc  2  0
2. 2 cos  3  0
3.
5. 5sec  10  0
6. 4 cos 2   1
7. cos  2  3cos
8. tan 2   tan   0
9. 2 cos 2   5 cos  2  0
10. sec  2  0
11. 5sin   13  3sin   14
12.
4.
2 cos  1  0
3 cot  1  0
Extra Credit: Solve the following equations on the interval 0    2 . Give the exact answer in terms of  .
tan  sec  tan   0
14