Download Pre-Calculus Unit 7- 2nd term November 17th – December 17th 2010

Pre-Calculus Unit 7- 2nd term November 26th – December 21st 2012
Chapter 5:Analytic Trigonometry and Semester Review/Exams
Date
Monday 11/26
Tuesday 11/27
Wednesday 11/28
Thursday 11/29
Friday 11/30
Monday 12/3
Tuesday 12/4
Wednesday 12/5
Thursday 12/6
Friday 12/7
Monday 12/10
Tuesday 12/11
Wednesday 12/12
Thursday 12/13
Friday 12/14
Monday 12/17
Tuesday 12/18
Wednesday 12/19
Thursday 12/20
Friday 12/21
Topic
Trig Applications and inverse trig
Trig Applications and inverse trig
Test Chap 4
Review
Test #7: Chapter 5
Fall Final Review
Fall Final Review
Fall Final Review
Semester Exam (6th)
Semester Exams (2nd, 4th )
Semester Exams (3rd , 5th)
Semester Exams ( 1st , 7th )
Go over Properties.
1  tan 2   sec 2 
sin 2   cos 2   1
10.
csc 2 x  1
cot x
3. (1  cos x)(1  cos x)
2. sec x cot x
6. sec x (sec x  cos x )
Simplify the following.
Memorize these:
Pythagorean Identities:
1. tan x cos x
WS
Worksheet page 2
Worksheet page 3
Worksheet page 4
Worksheet page 4
Worksheet page 5
Worksheet page 5
Worksheet page 6
Study for test
Work on fall final review
Work on fall final review
Work on fall final review
Study for your exams
Study for your exams
Study for your exams
Study for your exams
Have a great Break 
5.1 Trig Identities
5.1 Trig Identities
5.2 Verifying Trig Identities
5.2 Verifying Trig Identities
5.3 Solving trig equations
5.3 Solving trig equations
5.3 Solving trig equations
Notes: 5.1 Trigonometric Identities
Did it
Assignment
WS evens
Study for test
7. cot x  csc x
2
11.
cos x sin x
1  sin 2 x
2
4. (sec x  1)(sec x  1)
8. (csc A  1)(csc A  1)
12.
1  cot 2   csc 2 
9.
5. sin x  1
2
sin   cot  cos 
sin x cos x

csc x sec x
page 1
Assignment 5.1 Simplify the following.
1. sec2 θ (1 – sin2 θ)
6.
11.
sec x cot x sin x
2.
7. 1  cot A
1
1

2
cos A cot 2 A
2
12.
15. (sec B  tan B)(sec B  tan B )
19.
3. (sin   1)(sin   1)
sin x cot x
(sec2 x  1)(csc2 x  1)
23. (1 – sin2θ)(1 + tan2θ)
8.
csc2 x(1  cos 2 x)
sin 2 a
1  cos a
16.
sin x cos x
1  cos 2 x
4. 1  tan
9.

5. tan x  sec x
2
1
1

2
sin A tan 2 A
13. cos x(sec x  cos x)
17.
2
tan 2 x
1
sec x  1
20. tan2 θ – sec2θ
21. tan csc
24. secθ – (tanθ sinθ)
25. cos (sec – cos)
10. 1 
14.
cos2 A(sec2 A  1)
18.
sin 2 x
+ cos x
cos x
22.
2
1  cot x 1  cos x
2
2
26. csc2x (1 – cos2x)
page2
sin 2 A
tan 2 A
Assignment
5.1 Simplify the following.
1. sin (–x)
2. cos (–x)
8.
13.
1  tan 2 x
1  cot 2 x
sec  tan 
–
cos  cot 
9.
3. tan (–x)
tan 2 x  1
tan x sec 2 x
14.
sec x  tan x sec x  tan x
21.
sec x cos x  sin 2 x sec x

10.
csc 2 x  1
cot 2 x
 cos 2   sin 2    1 

 

cos 

  sec  
17.

4. csc (–x)
18. sinθ (cscθ – sinθ)
22.
sin x tan x  cos x
5. sec (–x)
11.
15.
7. (cos2θ)(sec2θ–1)
6. cot (–x)
sec 2 x  tan 2 x
csc x
12.
 cos 2 x 

 + sin x
 sin x 
19. cotx secx
23. sec x  sin
16.
1
1
+
2
sec x csc 2 x
sec 2 x
sec 2 x  1
20. 2cos2 θ – sin2 θ + 1
x tan x
24. tanx (sinx + cosx cotx)
Page3
Assignment 5.2 Verifying Trig Identities
Verify each identity. Work only on one side.
1  tan 2 x
 sec 2 x
2
2
sin x  cos x
1.
1
1

1
sec 2  csc 2 
2.
4.
sec 2 x(1  sin 2 x)  1
5. sin x(sec x  csc x)  tan x  1
7.
sec x
 sin x
tan x  cot x
8.
sin x  cos x cot x  csc x
3.
1  tan  1  sin    1
2
2
6. cos x csc x  cot x
9.
1
 tan   sec  csc 
tan 
10.
sin 2 x
 cos x  1
1  cos x
11.
1  tan 2 
 csc 2 
2
tan 
12.
1  sin 2 
 cot 2 
2
1  cos 
13.
sin x cot x  cos x
 2 cot x
sin x
14.
sin x  cos x
 1  tan x
cos x
15.
sin 2 x
 sec 2 x  1
2
cos x
16. 2 cos x  sin x  cos x  1
2
19.
2
sec 
 sin 
tan   cot 
2
17. sin
2
  cos 2   1  2 cos 2 
20. 1  2 sin
2
x  2 cos 2 x  1
18.
sin x
 cos x
tan x
21. 1  cot
2
  csc 2 
page 4
Notes/ Homework
I
5.3 Trig Equations
Solve each equation.
Collecting Like Terms
1.
sin x  2   sin x
4.
2 cos x  3  0
II
2
3 sec x  2  0
6.
3.
sin x  3  3 sin x
4 cos x  2  cos x  1
2 sec x  2
7.
9. 3 sec x  4
10. 3 cos x  3  cos x
2
2
2
Factoring out GCF
11. cot x cos x  cot x
2
15.
5.
5 cos x  3  3 cos x  4
Extracting Square Roots
8. 3 tan x  1  0
III
2.
3 tan x sin x  sin x
12. csc x  4 csc x  0
4
16.
2
13. sin
2 sin x cos x  2 sin x  0
2
x  sin x  0
14. 2 cos x  cos x
2
17. tan x sec x  tan x
page 5
IV
Factoring an equation of Quadratic Type
18. 2 sin x  sin x  1  0
19.
21. tan x  2 tan x  1  0
22. sec x  3 sec x  2
2
2
Functions of Multiple Angles
24.
cos 3x  1
VI
Substituting using Formulas:
32. sin
2
20. sec x  sec x  2  0
2
23. 2 cos x  3 cos x  1  0
2
V.
25. tan 3 x  3
29. 2 sin x  3 cos x  3
2 sin 2 x  5 sin x  3
2
26. 2 sin 2 x  1
2
2
27.
sin 2x  1
28.
sec 2x  2
2
30. 2 sec x  2 tan  4  0
31. 2 cos x  2 sin
x  cos 2 x  1
33. 2 sin x  2  cos x
34. 2 sec x  tan x  3
2
2
2
2
2
2
x 1
2
page 6
Precalc 5.4 Sum and Difference Formulas
sin(    )  sin  cos   cos  sin 
tan(    ) 
tan   tan 
1  tan  tan 
tan(    ) 
tan   tan 
1  tan  tan 
sin(    )  sin  cos   cos  sin 
cos(   )  cos  cos   sin  sin 
cos(   )  cos  cos   sin  sin 
Simplify each Expression.


 sin

_________ 1. cos25°cos15° – sin25°sin15°
_________ 2. cos
_________ 3. sin140°cos50° – cos140°sin50°
_________ 4. sin3cos1.2 – cos3sin1.2
7
cos
5
7
sin
_________ 5.
tan 325  tan 86
1  tan 325 tan 86
_________ 6.
tan 2 x  tan x
1  tan 2 x tan x
_________ 7.
tan 140  tan 60
1  tan 140 tan 60
_________ 8.
tan 240  tan 140
1  tan 240 tan 140

5
Evaluate:
______________ 9. sin75°cos15° + sin15°cos75°
_______________ 10. sin15°cos30° + sin30°cos15°
______________ 11. cos105°cos60° + sin 105°sin60°
______________ 12. cos105°cos15° + sin 105°sin15°
______________ 13.
tan 75  tan 30
1  tan 75 tan 30
______________ 14.
tan 60  tan 30
1  tan 60 tan 30
______________ 15.
tan 100  tan 50
1  tan 100 tan 50
______________ 16.
tan 200  tan 70
1  tan 200 tan 70
______________ 17. cos
5

5

cos  sin
sin
12
12
12
12
______________ 18. cos
7
5
7
5
cos
 sin
sin
6
6
6
6
______________ 19. sin
4


4
cos  sin cos
3
3
3
3
______________ 20. sin
2


2
cos  sin cos
3
3
3
3
Find the exact value of each expression. (no calculators)
21. cos 75°
22. cos 165°
23. sin 105°
24. sin 75°
page 7
Evaluate.
25. If sin x 
3
24
(x is in Quadrant I) and sin y 
(y is in Quadrant II), find sin (x + y).
5
25
26. If sin x 
4
1
(x is in Quadrant III) and sin y 
(y is in Quadrant II), find cos (x – y).
5
2
Given sin u 
5
3
and cos v 
, both in quadrant II. find:
13
5
27. sin (u + v)
28. cos (u + v)
29. tan (u + v)
30. sin (u – v)
31. cos (u – v)
32. tan (u – v)
page 8
Pre Calc 5.5 Double Angle Formulas
cos 2 A  cos 2 A  sin 2 A
sin 2 A  2 sin A cos A
tan 2 A 
2 tan A
1  tan 2 A
cos 2 A  2 cos 2 A  1
cos 2 A  1  2 sin 2 A
Simplify:
2 cos 2 10  1
x
x
2 sin cos
2
2
__________ 2) 2 cos 2 42  1
x
x
__________ 4) 2 sin cos
4
4
_________ 5)
2 tan 3x
1  tan 2 3x
__________ 6)
2 tan 16
1  tan 2 16
_________ 7)
cos 2 4 A  sin 2 4 A
__________ 8)
cos 2 13  sin 2 13
_________ 9)
1  2 sin 2 21
__________ 10)
1  2 sin 2 49
_________ 11)
2 tan 25
1  tan 2 25
__________ 12)
1  2 sin 2
_________ 13)
4 tan x
1  tan 2 x
__________ 14)
4 sin
_________ 1)
_________ 3)
x
2
7
7
cos
12
12
Simplify, then Evaluate:
______________ 15)
2 sin 15 cos15
_______________ 16)
______________ 17)
2 tan  / 8
1  tan 2  / 8
_______________ 18)
______________ 19)
cos 2
______________ 21)
1  2 sin 2 45
______________ 23)
cos 2
______________ 25)
2 sin 90 cos 90

12

3
 sin 2
 sin 2

12

3
2 cos 2  / 8  1
1  2 sin 2
_______________ 20) 4 sin

8

12
cos

8
_______________ 22)
2 sin 30 cos 30
_______________ 24)
6 tan 75
1  tan 2 75
_______________ 26)
2 cos 2  / 12  1
page 9
Pre Calc 5.5 Half Angle Formulas
sin
A
1  cos A

2
2
cos
A
1  cos A

2
2
tan
A
1  cos A

2
1  cos A
tan
A 1  cos A

2
sin A
tan
A
sin A

2 1  cos A
Simplify:
1  cos 80
2
__________ 1)
__________ 3) 
1  cos  / 7
2
1  cos  / 6
2
__________ 5)
__________ 7) 
1  cos 32
2
__________ 2)
__________ 4) 
__________ 6)
__________ 8) 
1  cos14
2
1  cos  / 5
2
1  cos  / 4
2
1  cos 212
2
__________ 9)
1  cos 26
sin 26
__________ 10)
1  cos 64
sin 64
__________ 11)
1  cos  / 7
sin  / 7
__________ 12)
1  cos  / 9
sin  / 9
__________ 13)
__________ 15)
1  cos 44
1  cos 44
sin 66
1  cos 66
__________ 14)
1  cos 74
1  cos 74
__________ 16)
sin 106
1  cos 106
Simplify, then Evaluate:
_______________ 17)
_______________ 19)
1  cos  / 2
sin  / 2
1  cos 240
2
_______________ 18)
_______________ 20)
sin 120
1  cos 120
1  cos  / 3
2
page 10