Honors Pre-Calculus
Chapter 6
Analytic Trigonometry
Mrs. Boddy
Chapter 6 Assignments
2/22
F:
p. 468 1-11 o, 33,35 p. 474 1-45 eoo
2/25
M:
p. 480 5-23 o
T:
p. 481 25,27,31,37,39,43,47,53
W: p. 481 40,50,57,59,63,66,67
3/4
R:
p. 491 3-19 eoo, 25, 29-41 eoo
F:
p. 492 31-43 eoo, 57, 58, 61, 65
M:
p. 501 1-21 eoo, 29, 33, 41, 53, 57 due Wednesday
T:
6.1-6.4 QUIZ (NO CALC)
W: p. 511 11,15,27,31,33,35,39
3/11
R:
p. 511 13,19,29,37,41 p. 518 1-11 o
F:
p. 518 13-19 o, 25-33 o
M:
Review
T:
Review
W: Ch 6 TEST
R:
Qtr Review
F:
QTR EXAM
Honors Pre-Calculus
6.1/6.2
Inverse Trig Functions
Learning Targets: Students will be able to find the exact and approximate value of the inverse sine, cosine,
and tangent functions.
Review:
To test for a function: Vertical Line Test
To test for a one-to-one function: Horizontal Line Test
Let’s look at 4 trig graphs, and decide where the graphs would be a one-to-one function.
Sine (Cosecant will be the same)


Tangent

Cosine (Secant will be the same)
2

2
Cotangent



Now, we are going to look at the inverse trig functions. Inverse trig functions can be found in the part of the
wave that is one-to-one. The above discussion will now tell us where we can find our solutions to the inverse
functions.
Find the value of  for the following problems. (Find the value of  where this will work).
(Remember that when working with right triangle trig, the inverses found angle measures…this is the same
idea, but in RADIANS)
 3
2. cos1 1
8. tan 1 

 3 

2
12. sin 1  

 2 
  2
34. Does sin 1  sin 
  3
  2
?
 
 3
Use the expression in the parentheses to draw the proper triangle, and then solve for the outer most trig value.

 1 
2. sin  cos 1   
 2 



3 
10. cos  sin 1  
 

 2 


 1 
18. tan  cos 1   
 3 

 2 3
36. csc 1  

3 

On the calculator:
38. csc1 5
40. sec1  3
Honors Pre-Calculus
6.3 Day 1
Trig Identities
Learning Targets: Students will be able to verify trigonometric identities.
Reciprocal Identities:
csc  
1
sin 
sec  
1
cos 
cot  
cos 
sin 
cot  
1
tan 
Quotient Identities:
tan  
sin 
cos 
Pythagorean Identities:
sin 2   cos 2   1
tan 2   1  sec2 
1  cot 2   csc2 
sin 2 
tan  
cos 2 
1
csc 2  
sin 2 
cos 2 
cot  
sin 2 
Variations:
sin   1  cos 
2
2
cos 2   1  sin 2 
2
2
sec 2  
1
cos 2 
cot 2  
Hints:
1. Start on the side that has the most changes to make (looks the hardest).
2. Change everything into sin and cos.
3. Use all algebra skills (and work ONE side only).
Ex.
tan 
 sin 
sec 
 sec  tan  
2

1  sin 
1  sin 
1
tan 2 
Ex.
sin 4 x  cos 4 x  2sin 2 x  1
Honors Pre-Calculus
6.3 Day 2
Trig Identities
Learning Targets: Students will be able to verify trigonometric identities.
Reciprocal Identities:
csc  
1
sin 
sec  
1
cos 
cot  
cos 
sin 
cot  
1
tan 
Quotient Identities:
tan  
sin 
cos 
Pythagorean Identities:
sin 2   cos 2   1
sin 2   1  cos 2 
cos 2   1  sin 2 
tan 2   1  sec2 
1  cot 2   csc2 
Hints:
1. Start on the side that has the most changes to make (looks the hardest).
2. Change everything into sin and cos.
3. Use all algebra skills (and work ONE side only).
Ex.
csc   1 1  sin 

csc   1 1  sin 
tan   cot 
 2 cos 2   1
tan   cot 
Ex.
cot 
tan 

 1  tan   cot 
1  tan  1  cot 
Honors Pre-Calculus
6.4 Day 1
Sum and Difference Formulas
Learning Targets: Students will be able to use the sum and difference formulas to verify identities.
How do we deal with angles that are not one of the special angles?
30
 60  90 or 45  45  90

The following formulas will help us solve the problems: (These formulas need to be MEMORIZED!)
cos(   )  cos  cos   sin  sin 
cos(   )  cos  cos   sin  sin 
sin(   )  sin  cos   cos  sin 
sin(   )  sin  cos   cos  sin 
tan(   ) 
tan   tan 
1  tan  tan 
tan(   ) 
tan   tan 
1  tan  tan 
Ex) cos

12
14.
sin 20 cos80  cos 20 sin 80
26. tan  
5
3
1
3
,   
; sin    ,    
12
2
2
2
a. sin(   ) 
b. cos(   ) 
c. sin(   ) 
d. tan(   ) 
30. If cos  
c. cos( 
1
,   QIV , find the exact value.
4

)=
3
38. Prove
tan(2   )   tan 
Honors Pre-Calculus
6.4 Day 2
Sum and Difference Formulas
Learning Targets: Students will be able to use the sum and difference formulas to verify identities.
How do we deal with angles that are not one of the special angles?
30
 60  90 or 45  45  90

The following formulas will help us solve the problems: (These formulas need to be MEMORIZED!)
cos(   )  cos  cos   sin  sin 
cos(   )  cos  cos   sin  sin 
sin(   )  sin  cos   cos  sin 
sin(   )  sin  cos   cos  sin 
Ex.
tan(   ) 
tan   tan 
1  tan  tan 
tan(   ) 
tan   tan 
1  tan  tan 
3 

sin  cos 1  
5 6

62.

5
 3 
cos  tan 1  sin 1    
12
 5 

Honors Pre-Calculus
6.5 Day 1
Double & Half Angles
Learning Targets: Students will be able to use double and half-angle formulas to verify identities.
Double Angle Formulas
sin  2   2sin  cos 
cos 2   sin 2 

cos  2   2cos 2   1
1  2sin 2 

tan  2  
1  cos 
 
cos    
2
2
1  cos 
 
tan    
1  cos 
2
2 tan 
1  tan 2 
Half Angle Formulas
1  cos 
 
sin    
2
2
To determine the  , you decide based on the location of  (the original angle).
Examples:
6. sin   
Find:
3 3
,
   2
3
2
a.
sin  2 
b.
cos  2 
c.
 
sin  
2
d.
 
cos  
2
 9 
16. tan 

 8 
Verify the following using double or half angles.
1
34. csc  2   sec  csc 
2
2
 
40. csc 2   
 2  1  cos 
Honors Pre-Calculus
6.7
Trig Equations
Learning Targets: Students will be able to solve trigonometric equations.
To solve these equations, we will be using all of the trig skills that we have been working with in the past few
weeks.
Remember:
1. All of the algebra skills we have practiced (Factor, Foil, Quadratic Equation)
2. Try to convert to one trig function (may not work for all)
3. Never divide by a trig function.
4. If stuck, square both sides and check solutions.
Solve the following equations in the interval 0    2 .
4. cos   
3
2
Ex. sin  2   0
16. 4 cos 2   3  0
32. 5csc  3  2
On the calculator, find the following.
36. cos  0.6
40. sin   0.2
Honors Pre-Calculus
6.8
Trig Equations
Learning Targets: Students will be able to solve trigonometric equations.
Remember:
1. All of the algebra skills we have practiced (Factor, Foil, Quadratic Equation)
2. Try to convert to one trig function (may not work for all)
3. Never divide by a trig function.
4. If stuck, square both sides and check solutions.
Solve the following equations in the interval 0    2 .
4. 2 cos 2   cos   1  0
10. 2sin 2   3 1  cos 
14. cos  sin   0
28. csc2   cot   1
16. sin  2   cos