Download Lesson Plans—Algebra I

PRE-AP PRECALCULUS
Fundamental Trigonometric Identities
Reciprocal Identities:
Quotient Identities:
sin u 
1
csc u
cos u 
1
sec u
tan u 
1
cot u
tan u 
sin u
, cos u  0
cos u
csc u 
1
sin u
sec u 
1
cos u
cot u 
1
tan u
cot u 
cos u
, sin u  0
sin u
Pythagorean Identities:
sin 2 u  c o s2 u 1
1  tan 2 u  sec2 u
1  cot 2 u  csc2 u
Odd/Even Identities:
sin(u )   sin(u )
csc(u )   csc(u )
cos(u )  cos(u )
tan(u )   tan(u )
cot(u )   cot(u )
sec(u )  sec(u )
Cofunction Identities:


sin   u   cos u
2



tan   u   cot u
2



cos   u   sin u
2



cot   u   tan u
2



sec   u   csc u
2



csc   u   sec u
2

1
Notes on Fundamental Trig Identities
Pythagorean Identities:
sin 2   cos 2   1
1  tan 2   sec 2 
1  cot 2   csc 2 
Use the fundamental identities to reduce as much as possible.
1. sinx cscx =
4.
sin 2 x  cos 2 x

sin x  cos x
7. cscx – cosx cotx
10. Rewrite
2.
sin x

cos x tan x
3. tanx cscx cosx =
5.
1  cos 2 x

sin x
6.
sin(  x )

cos(  x )
8.
1
1

1  sin x 1  sin x
9.
sin x
cos x

1  cos x sin x
1
so that it is not in fractional form.
1  sin x
11. Rewrite
cos 2 x
so that it is not in fractional form.
1  sin x
2
Assignment: Trigonometric Identities
I. Simplify the following. Show all work on your own paper as needed.
1. (1  cos  )(1  cos  )
2. sin x cot x
3. (sin   1)(sin   1)
4. 1  tan 2 
5. tan 2 x  sec2 x
6. sec x cot x sin x
7. 1  cot 2 A
8.
10. 1 
sin 2 A
tan 2 A
13. cos x(sec x  cos x)
11.
sin 2 a
1  cos a
9.
1
1

2
cos A cot 2 A
14. cos2 A(sec2 A  1)
sec 2 x  tan 2 x
csc x
12. csc2 x(1  cos 2 x)
15. (sec B  tan B)(sec B  tan B)
17.
tan x  cot x
sec 2 x
18. (sin x  cos x)2  (sin x  cos x)2
19. (sec2 x  1)(csc2 x  1)
20.
cot 2 x
 sin x csc x
1  csc x
21.
22. cos3 y  cos y sin 2 y
23.
sec y  csc y
1  tan y
24. (csc A  1)(csc A  1)
25. sin   cot  cos
26. 2cos2 θ – sin2 θ + 1
16.
28.
sin x cos x
1  cos 2 x
cos x 1  sin x

1  sin x
cos x
*29. Rewrite
27.
tan 2 x
1
sec x  1
sec x sin x

sin x cos x
5
so that it is not in fractional form.
tan x  sec x
3
Assignment: Trigonometric Identities
Simplify. Show all work on your own paper.
1. sin(-x)
2. cos(-x)
3. tan(-x)
7. tanx (sinx + cosx cotx)
10.
csc 2 x  1
cot 2 x
4. csc(-x)
5. sec(-x)
6. cot(-x)
8. (1+ tan2x)(1-sin2x)
11.
sec 2 x  tan 2 x
csc x
9. cotx secx
12.
1
1
+
2
sec x
csc 2 x
13. tan2 θ - sec2θ
14. sec2 θ (1 – sin2 θ)
15. 2cos2 θ – sin2 θ + 1
 cos 2   sin 2    1 
 
16. 

cos 

  sec  
 cot  
17. 
 + tan 
 csc   1 
18.
20. sin x tan x  cos x
20. sec x  sin x tan x
22 sec x  tan x  sec x  tan x
23. sec x cos x  sin 2 x sec x
19.
tan 2 x  1
tan x csc 2 x
21. cos xtan x  cos x
24.
sin   tan 
1  sec 
25.
 cos 2 x 
 + sin x
27. 
 sin x 
sec 
tan 
–
cos 
cot 
sec x sin x
tan x  cot x

26.
2  cot 2 
–1
csc 2 
1  tan 2 x
28.
1  cot 2 x
sin 2 x
29.
+ cos x
cos x
31. (cos2θ)(sec2θ–1)
32. sinθ (cscθ – sinθ)
33. (1 – sin2θ)(1 + tan2θ)
34. secθ – (tanθ sinθ)
35. cos(sec – cos)
36. csc2x (1 – cos2x)
37. tan csc
38. cot2 - (cos4 csc4)


30. 1  cot 2 x 1  cos 2 x


4
Notes on Verifying Trig Identities
Guidelines:
1.
2.
3.
4.
5.
6.
Work with one side at a time – usually the most complicated side
Tools to consider: Factoring, adding fractions, squaring binomials
May need to transform the denominator so…transform it to a monomial
Use the fundamental identities
Try converting all terms to sines and cosines
Always try something!
Verify the identity:
1.
sin 2 x  cos 2 x
1
cos 2 x sec 2 x
2.
1
1

 2 csc 2 x
1  cos x 1  cos x
3. (sec2x – 1) (sin2x – 1) = - sin2x
4. cscx – sinx = cosx cotx
sin x
5. cscx + cotx =
1  cos x
tan 2 x
1  cos x
6.

1  sec x
cos x
7. tan3x = tanx sec2x – tanx
8. cos3x sin4x = (sin4x – sin6x) cosx
5
Assignment: Simplifying and Verifying Identities
Simplify each of the following to sin(x), cos(x), tan(x), or 1
1.
1
1

2
sec x csc 2 x
2.
sec x
tan x  cot x
3.
5. (1 + tan2x)(1 – sin2x)
1  tan x
1  cot x
4. sinxsecx
6. secx – sinxtanx
Verify each identity. Work with one side only. Use separate paper as necessary.
1. sinx + cosxcotx = cscx
2. cosx cscx = cotx
4. sinx(secx – cscx) = tanx – 1
7.
10.
sin x cot x  cos x
 2 cot x
sin x
sec x
= sinx
tan x  cot x
13. cos x  sin x2  (cosx  sin x)2  2
3. 2cos2x – sin2x + 1 = 3cos2x
5.
1
1

 2 sec 2 x
1  sin x 1  sin x
6.
sin 2 x
 cos x  1
1  cos x
8.
sin x cos x
1

2
1  2 sin x cot x  tan x
9.
1  tan 2 x
 csc 2 x
tan 2 x
11.
1
 tan   sec  csc 
tan 
14.
12. sec 2 x(1  sin 2 x)  1
1  sin 2 
 cot2 
2
1  cos 
6
Notes on Solving Trig Equations
Solve each of the following for 0  x  2 .
1. sinx -
2 = -sinx
2. 4sin2x – 3 = 0
3. sin2x = 2sinx
4. 2sin2x – 3sinx + 1 = 0
5. 3sec2x – 2tan2x – 4 = 0
6. sinx + 1 = cosx
7. sin2x -
8. tan
x
1  0
2
3
0
2
9. 4tan2x + 5tanx = 6
7
Assignment: Solving Trig Equations
1. 2 cos x  3  0
3 sec x  2  0
2.
5. 4 cos x  2  cos x  1
8. cos 3x  1
Use separate paper as necessary.
6.
9. 2 tan 5x  2
2 sec x  2
10. sec 2x  2
3. sin 2x  1
7.
11.
4. tan 2x  1
3 tan x sin x  sin x
2 sin x cos x  2 sin x  0
12. sin 2 x  sin x  2  0
13. 2 cos 2 x  5 cos x  3  0
14. tan 2 x  2 tan x  1  0
15. sec 2 x  3 sec x  2
16. 4 sin 2 x  4 sin x  1  0
17. 2 cos 2 x  3 cos x  1  0
8
18. 2 cos 2 x  cos x
19. tan x sec x  tan x
20. 2 sin 2 5 x  3 sin 5 x  1  0
21. sin 2x  cos x
22. sin 2x  2 cos x  0
23. cos 2x  sin x  0
24. cos 2x  sin x  1
25. cos2x  3cosx  1
26. cos 2x  cos x
27.
1  cos x
 1
sin x
30. 2 sin 2 x  2  cos x
28.
sin 2 x
0
1  cos 2 x
31. 2 sin x  csc x  0
29. sin 2 x  cos 2 x  1
32. 2 sec 2 x  tan 2 x  3
9
Notes on Sum/Difference Identities
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 
Evaluate using sum and difference formulas – (expanding) – to find the exact value of the following:
13
1. sin 15◦
2. cos
1. “split” the angle
12
2. Write the formula
3. Fill in the values
4. Simplify
Evaluate or simplify using sum and difference formulas – (compressing) – find the exact value of …
3
2
3
2
1. cos25◦ cos20◦ - sin25◦ sin20◦
2. sin
cos
-- cos
sin
5
5
5
5
Extra Problems:
1. Write sin(arctan1 + arccosx) as an algebraic expression
10


2. Prove: sin  x     cos x
2




5. Simplify: cos 30  x  cos 30  x
 3

 x
3. Simplify: sin 
 2




4. Simplify: tan  x  
4


3 


6. Solve: sin  x    sin  x 
 1
2
2 


on [0, 2π)
3
5
and tan   
where  lies in quadrant II and  lies in quadrant IV. Show your work for
5
12
the following. Express your answers as lowest term fractions.
Given sin  
a. cos    
b. sin    
c. tan    
11
Assignment: Sum and Difference Identities
Simplify each Expression.
1. cos 42cos18  sin 42sin18
2. sin120cos 25  cos120sin 25

2

2
 sin cos
3. cos cos
7
3
7
3
4. sin
5. sin3cos1.2 – cos3sin1.2
6.
tan 240  tan 140
7.
1  tan 240 tan 140
3
2
3
2
cos
 cos sin
8
5
8
5
tan 2 x  tan x
1  tan 2 x tan x
tan   tan
8.

1  tan  tan
6

6
9. cos 71cos 29  sin 71sin 29
Evaluate the following.
10. sin15°cos30° + sin30°cos15°
11. cos105°cos15° + sin 105°sin15°
12
12.
tan 60  tan 30
1  tan 60 tan 30
14. cos
13.
7
5
7
5
cos
 sin
sin
6
6
6
6
tan 200  tan 70
1  tan 200 tan 70
15. sin
16. sin 25cos 35  cos 25sin 35
17. cos105cos15  sin105cos15
Evaluate the exact value of each expression (No Calculators)
7
18. cos
19. sin 75°
12
21. cos
7
12
2


2
cos  sin cos
3
3
3
3
22. sin

12
20. cos(15)
23. tan 225
13
Simplify:


24. Simplify: tan  x  
4

26. Solve: sin( x 

3
)  sin( x 



25. cos 30  x  cos 30  x

3

)
Evaluate.
27. If sin x 
3
24
(x is in Quadrant I) and sin y 
(y is in Quadrant II), find cos (x + y).
5
25
28. Given sin u 
a. cos (u + v)
5
3
and cos v 
, both in quadrant II. Find.
13
5
b. sin (u – v)
c. tan (u – v)
14
Notes on Double Angle Trig Identities
5.5 Double Angle Formulas
sin 2 A  2 sin A cos A
tan 2 A 
2 tan A
1  tan 2 A
cos 2 A  cos 2 A  sin 2 A
cos 2 A  2 cos 2 A  1
cos 2 A  1  2 sin 2 A
3
5
and tan   
where  lies in quadrant II and  lies in quadrant IV.
5
12
Show your work for the following. Express your answers as lowest term fractions.
Given sin  
1) sin 2   _______________
2) tan2   _______________
3) cos2   _______________
II. Express each in terms of the sine, cosine, or tangent of a single angle.
4)
tan 60  tan 45
1 tan 60  tan 45
 
 
6) 2 sin   cos 
9
9

5) 1 2sin 2  
 12 
7)
2 tan 45
1  tan 2 45
15
Simplify, then Evaluate:

8) 2 cos 2    1
 6
III.
Solve the following equations over the interval 0, 2 
10) sin 2x  cos x  0
12)
9. cos 2 60  sin 2 60
11) tan( x   )  2sin( x   )  0
cos2x  cos x  0
16
Notes on Half Angle Trig Identities
Half Angle Formulas:
A
1  cos A

2
2
A
1  cos A
tan  
2
1  cos A
sin
cos
A
1  cos A

2
2
tan
A 1  cos A

2
sin A
tan
A
sin A

2 1  cos A
3
5
and tan   
where  lies in quadrant II and  lies in quadrant IV. Show your work
5
12
for the following. Express your answers as lowest term fractions.
Given sin  
 
1) sin    _______________
2
 
2) tan    _______________
2
 
3) cos    _______________
2
II. Use half angle identities to evaluate. Write radicals in simplest form.
 
 3 
4) sin  
5) cos 165 
6) tan 

8
 8 
2
3
7)
2
1  cos
3
sin
5
3
8) 
5
1  cos
3
1  cos
 7 
9) sec 

 12 
17
Assignment Double/Half Angle Trig Identities
Simplify:
_________ 1)
_________ 3)
2 cos 2 10  1
_________ 2) 2 sin
2 tan 3x
1  tan 2 3x
_________ 4) cos 2 4 A  sin 2 4 A
_________ 5) 1  2 sin 2 21
_________ 7)
x
x
cos
2
2
2 tan 25
1  tan 2 25
_________ 6)
4 tan x
1  tan 2 x
__________ 8) 4 sin
7
7
cos
12
12
Simplify, then Evaluate:
_________ 9) 2 sin 15 cos15
_________ 11) cos 2

12
 sin 2
_________ 10)

2 tan  / 8
1  tan 2  / 8
_________ 12) 1  2 sin 2 45
12
_________ 13) 2 cos 2  / 12  1
_________ 14) 2 sin 30 cos 30
Find the exact Solution of the equations on the interval 0  x  2 . Show all work.
15) sin 2x   cos x
16. cos 2x  sin x  0
17. tan 2x  2 cos x  0
18