Trigonometry Sec. 05 notes
MathHands.com M´ arquez
Famous IDs: Pythagoras Identities Main Idea The previous section was extremely important for a few reasons. First, you should now be comfortable understanding what an identity is. Second, we saw methods used to establish identities, working on each side independently, tweaking a known identity, graphing the functions and comparing the graphs, and last, proving identities by starting with a very creative idea. Moreover, we introduced the first dose of fundamental and famous trigonometric identities, most or all of which were proved in the corresponding assignment. Now, the task at hand is to expand the list of very famous trigonometric identities, and to practice our proving skills. Below is our next dose of famous identities, the first of which is the most famous of them all, the pythagoras identities. Pythagoras Identities sin2 θ + cos2 θ = 1
tan2 θ + 1 = sec2 θ
sin2 θ = 1 − cos2 θ
cos2 θ = 1 − sin2 θ
tan2 θ = sec2 θ − 1
cot2 θ + 1 = csc2 θ p sin θ = ± 1 − cos2 θ p sec θ = ± tan2 θ + 1
cot2 θ = csc2 θ − 1 p tan θ = ± sec2 θ − 1
EXAMPLE 1 (work on one side) The first identity in this family is the most famous trigonometric identity. We will prove it by working on one side. Without loss of generality, we can assume the following is a reference triangle for the angle θ c
b
θ a
sin2 θ + cos2 θ b 2 c b2 c2
+
+
b2 +a2 c2 c2 c2
1
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a2 c2
a 2 c
? = | | | | | | | | | =
1
want to know definitions of sine and cosine algebra clean up algebra clean up Pythagoras Theorem, see reference triangle
1
algebra clean up
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Trigonometry Sec. 05 notes
MathHands.com M´ arquez
EXAMPLE 2 (tweak a known identity) Prove the following is an identity. tan2 θ + 1 = sec2 θ Solution: We will prove this identity by tweaking the previous one, the Pythagoras Identity. sin2 θ + cos2 sin2 θ + cos2
1 cos2 θ sin2 θ cos2 θ
= 1
+
sin θ 2 cos θ
cos2 θ cos2 θ
+
2
cos θ 2 cos θ 2
We already know this one, proven above!
=
1 cos2 θ
=
1 cos2 θ
=
(1)
algebra
2 1 cos θ
(tan θ) + (1)
= (sec θ)
tan2 θ + 1
= sec2 θ
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multiply both sides
2
algebra proven identity done!
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Trigonometry Sec. 05 exercises
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Famous IDs: Pythagoras Identities
1. Prove and OWN [VERY famous identity]. sin2 θ + cos2 θ = 1 2. Prove and OWN [VERY famous identity]. tan2 θ + 1 = sec2 θ 3. Prove and OWN [VERY famous identity]. tan2 θ = sec2 θ − 1 4. Prove and OWN [VERY famous identity]. cot2 θ + 1 = csc2 θ 5. Prove and OWN [VERY famous identity]. cot2 θ = csc2 θ − 1 6. Prove the following famous identity sin2 θ = 1 − cos2 θ
7. Prove and OWN [VERY famous identity]. p sin θ = ± 1 − cos2 θ 8. Prove and OWN [VERY famous identity]. tan θ = ±
p sec2 θ − 1
9. Prove and OWN [VERY famous identity]. p sec θ = ± tan2 θ + 1 10. Prove the following famous identity tan2 θ = (sec θ − 1)(sec θ + 1)
11. Prove the following is an identity OR prove the following is not an identity 1 = sec2 θ + tan θ sec θ 1 − sin θ
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Trigonometry Sec. 05 exercises
MathHands.com M´ arquez
12. Prove the following is an identity OR prove the following is not an identity 1 = cot2 θ + cot2 θ sec θ 1 − sec θ
13. Prove the following is an identity OR prove the following is not an identity 1 1 + = 2 sec2 θ 1 − sin θ 1 + sin θ
14. Prove the following is an identity OR prove the following is not an identity cos4 θ = 1 − 2 sin2 + sin4 θ
15. Prove the following is an identity OR prove the following is not an identity cos6 θ = 1 − 3 sin2 θ + 3 sin4 θ − sin6 θ
16. Practice Work on each side: Assume there is an interesting world in which x2 + y 2 = 1 for all values of x and y. In such world, Determine if the following is an identity, if so answer TRUE, if not answer FALSE. 2x2 − 1 = 1 − 2y 2 A. TRUE
B. FALSE
17. Practice Work on each side: Determine if the following is an identity, prove your answer 2 cos2 x − 1 = 1 − 2 sin2 x A. TRUE
B. FALSE
18. Practice Work on each side: Assume there is an interesting world in which x2 + y 2 = 1 for all values of x and y. In such world, Determine if the following is an identity, if so answer TRUE, if not answer FALSE. (x + y)2 = 2xy + 1 A. TRUE
B. FALSE
Therefore TRUE 19. Practice Work on each side: Determine if the following is an identity, prove your answer. (cos x + sin x)2 = 2 cos x sin x + 1 A. TRUE
B. FALSE
Therefore TRUE
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Trigonometry Sec. 05 exercises
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20. Practice Work on each side: Determine if the following is an identity, prove your answer. cos4 x − sin4 x = 2 cos4 x − 2 cos2 x + 1 A. TRUE
B. FALSE
21. Practice Work on each side: Determine if the following is an identity, prove your answer. (cos x + sin x)4 = −4 cos4 x + 4 cos2 x + 4 cos x sin x + 1 A. TRUE
B. FALSE
22. Practice Work on each side: Determine if the following is an identity, prove your answer. 1 cos x + sin x
cos x − sin x = A. TRUE
B. FALSE
23. Practice Work on each side: Determine if the following is an identity, prove your answer. 1 2 cos x 1 + = cos x + sin x cos x − sin x 1 − 2 sin2 x A. TRUE
B. FALSE
24. Practice Work on each side: Determine if the following is an identity, prove your answer. 1 2 cos x 1 + = cos x + sin x cos x − sin x 2 cos2 x − 1 A. TRUE
B. FALSE
25. Practice Work on each side: Determine if the following is an identity, prove your answer. 1 1 = 2 2x−1 2 cos 1 − 2 sin x A. TRUE
B. FALSE
26. Practice Work on each side: Determine if the following is an identity, prove your answer. cos x − sin x = A. TRUE
1 2 cos x + sin x
B. FALSE
27. Practice Work on each side: Determine if the following is an identity, prove your answer. 1 cos x 1 − = cos x + sin x cos x − sin x 1 − 2 sin2 x A. TRUE
B. FALSE
28. Practice Work on each side: Determine if the following is an identity, prove your answer. 1 2 cos x 1 + = cos x + sin x cos x − sin x 1 − 2 cos2 x A. TRUE
B. FALSE
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Trigonometry Sec. 05 exercises
MathHands.com M´ arquez
29. Practice Work on each side: Determine if the following is an identity, prove your answer. cos6 x + 3 cos4 x sin2 x + 3 cos2 x sin4 x + sin6 x + 2 = 3 A. TRUE
B. FALSE
30. Practice Work on each side: Determine if the following is an identity, prove your answer. cos6 x + 3 cos4 x sin2 x + 3 cos2 x sin4 x + sin6 x − cos2 x = sin2 x A. TRUE
B. FALSE
31. Prove the following is an identity OR prove the following is not an identity cos6 θ = −2 + 3 cos2 θ + 3 sin4 θ − sin6 θ
32. Prove the following is an identity OR prove the following is not an identity 1 − cos4 θ = sin2 θ + sin2 θ cos2 θ
33. (**)Prove the identity sin(2x) = 2 sin x cos x
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math hands Trigonometry 05 exercises
Famous IDs: Pythagoras Identities
1. Prove and OWN [VERY famous identity]. sin2 θ + cos2 θ = 1
c
b
θ
Solution:
a
? = | | | | | | | | | =
sin2 θ + cos2 θ b 2 c b2 c2
a 2 c
+
+
a2 c2
b2 +a2 c2 c2 c2
1
1
want to know definitions of sine and cosine algebra clean up algebra clean up Pythagoras Theorem, see reference triangle
1
algebra clean up
2. Prove and OWN [VERY famous identity]. tan2 θ + 1 = sec2 θ
Solution: Solution: We will prove this identity by tweaking the previous one, the Pythagoras Identity. sin2 θ + cos2 1 cos2 θ sin2 θ cos2 θ
= 1
sin2 θ + cos2 +
sin θ 2 cos θ
cos2 θ cos2 θ
+
2
cos θ 2 cos θ 2
We already know this one, proven above!
=
1 cos2 θ
=
1 cos2 θ
=
(1)
multiply both sides algebra
2 1 cos θ
(tan θ) + (1)
= (sec θ)
tan2 θ + 1
= sec2 θ
2
algebra proven identities done!
3. Prove and OWN [VERY famous identity]. tan2 θ = sec2 θ − 1
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math hands Trigonometry 05 exercises Solution: by tweaking
tan2 θ + 1 = sec2 θ 2
(proven identity)
2
tan θ = sec θ − 1
(algebra)
4. Prove and OWN [VERY famous identity]. cot2 θ + 1 = csc2 θ
Solution: Solution: We will prove this identity by tweaking the previous one, the Pythagoras Identity. sin2 θ + cos2 1 sin2 θ sin2 θ sin2 θ
= 1
sin2 θ + cos2 +
sin θ 2 sin θ
cos2 θ sin2 θ
+
2
cos θ 2 sin θ 2
We already know this one, proven above!
=
1 sin2 θ
=
1 sin2 θ
=
(1)
multiply both sides algebra
2 1 sin θ
(1) + (cot θ)
= (csc θ)
1 + cot2 θ
= csc2 θ
algebra
2
proven identities done!
5. Prove and OWN [VERY famous identity]. cot2 θ = csc2 θ − 1
Solution: by tweaking
cot2 θ + 1 = csc2 θ 2
2
cot θ = csc θ − 1
(proven identity) (algebra)
6. Prove the following famous identity sin2 θ = 1 − cos2 θ
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math hands Trigonometry 05 exercises Solution: this is done by starting with a known (proven and famous) one sin2 θ + cos2 θ = 1. Then we isolate sin2 θ... then we are done!
sin2 θ + cos2 θ = 1
(famous & proven) 2
2
=1 − cos θ
(algebra, sub cos θ from both sides)
7. Prove and OWN [VERY famous identity]. p sin θ = ± 1 − cos2 θ
Solution: tweak....
sin2 θ = 1 − cos2 θ p sin θ = ± 1 − cos2 θ
(proven) (Square root property (algebra))
8. Prove and OWN [VERY famous identity]. tan θ = ±
p sec2 θ − 1
9. Prove and OWN [VERY famous identity]. p sec θ = ± tan2 θ + 1 10. Prove the following famous identity tan2 θ = (sec θ − 1)(sec θ + 1)
Solution: this is done by starting with a known (proven and famous) one tan2 θ = sec2 θ − 1. Then we factor the right side. then we are done!
tan2 θ = sec2 θ − 1 =(sec θ − 1)(sec θ + 1)
(famous & proven) (algebra, factor diff of squares)
11. Prove the following is an identity OR prove the following is not an identity 1 = sec2 θ + tan θ sec θ 1 − sin θ
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math hands Trigonometry 05 exercises
Solution: Famous method.... work on one side... I would start working on right side.. Famous idea.. SOMETIMES.... it helps to turn everything into sines and cosines.. 1 ? sec2 θ + tan θ sec θ = 1 − sin θ
=
(abandon =)
1 sin θ 1 + cos2 θ cos θ cos θ
=
(famous & proven ids)
sin θ 1 + cos2 θ cos2 θ
=
(algebra)
1 + sin θ cos2 θ
=
(algebra)
1 + sin θ 1 − sin2 θ
=
(famous & proven ids)
1 + sin θ (1 − sin θ)(1 + sin θ)
=
(algebra, factor)
1 1 − sin θ
=
(algebra, simplify, kachin, kachin, that is why they pay me)
=
12. Prove the following is an identity OR prove the following is not an identity 1 = cot2 θ + cot2 θ sec θ 1 − sec θ
Solution: this is not an identity
13. Prove the following is an identity OR prove the following is not an identity 1 1 + = 2 sec2 θ 1 − sin θ 1 + sin θ
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math hands Trigonometry 05 exercises
Solution: Famous method.... work on one side... I would start working on left side.. Famous idea.. SOMETIMES.... it helps to start on the side that looks more complex Another famous idea SOMETIMES.... its good to use conjugates, such as a-b and a+b 1 1 ? + = 1 − sin θ 1 + sin θ
2 sec2 θ
(abandon =)
1 1 + sin θ 1 1 − sin θ · + · 1 − sin θ 1 + sin θ 1 + sin θ 1 − sin θ
(conjugate idea, mult top and bottom)
1 + sin θ 1 − sin θ 2 + 1 − sin θ 1 − sin2 θ
(conjugates always clean up nicely)
1 + sin θ 1 − sin θ + cos2 θ cos2 θ
(famous & proven ids)
2 cos2 θ
(combine into one fraction)
2 sec2 θ=
(famous & proven ids)
14. Prove the following is an identity OR prove the following is not an identity cos4 θ = 1 − 2 sin2 + sin4 θ
Solution: Famous method.... work on one side... I would start working on left side.. Famous idea.. SOMETIMES.... it helps to start on the side that has higher exponents...
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math hands Trigonometry 05 exercises
cos4 θ
? 1 − 2 sin2 + sin4 θ =
(abandon =)
cos2 θ cos2 θ
(4th power is the same as square times squared, algebra)
(1 − sin2 θ)(1 − sin2 θ)
(famous & proven ids)
1 − 2 sin θ + sin4 θ=
(FOIL, from grades school algebra... kachin!)
15. Prove the following is an identity OR prove the following is not an identity cos6 θ = 1 − 3 sin2 θ + 3 sin4 θ − sin6 θ
Solution: Famous idea.. its no fun if the teacher give you all the answers..... you try this one.. try hard... start on the left side.. cos6 θ is cos2 θ · cos2 θ · cos2 θ etc... etc.. 16. Practice Work on each side: Assume there is an interesting world in which x2 + y 2 = 1 for all values of x and y. In such world, Determine if the following is an identity, if so answer TRUE, if not answer FALSE. 2x2 − 1 = 1 − 2y 2 A. True B. False Solution: 1 − 2y 2 2
(work on right side) 2
= x + y − 2y
2
(sub 1=x2 + y 2 )
= x2 − y 2 2
(algebra)
2
2
2
(clever! add zero)
2
(factor negative)
= x − y + −x + x 2
2
2
= x − (y + x ) + x 2
2
= x − (1) + x 2
= 2x − 1
(sub) (aglebra)
ALTERNATIVELY, observe x2 + y 2 = 1 implies x2 = 1 − y 2 , thus..
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math hands Trigonometry 05 exercises 1 − 2y 2
(work on right side) 2
= 1 − 2(1 − x )
(the tweaked sub)
2
= 2x − 1
(aglebra)
17. Practice Work on each side: Determine if the following is an identity, prove your answer 2 cos2 x − 1 = 1 − 2 sin2 x A. True B. False Solution: 1 − 2 sin2 x
(work on right side) 2
2
2
(sub 1=cos2 x + sin2 x)
= cos x + sin x − 2 sin x = cos2 x − sin2 x
(algebra)
2
2
2
2
(clever! add zero)
2
(factor negative)
= cos x − sin x + − cos x + cos x 2
2
2
= cos x − (sin x + cos x) + cos x 2
2
= cos x − (1) + cos x
(sub)
2
= 2 cos x − 1
(aglebra)
ALTERNATIVELY, observe cos2 x + sin2 x = 1 implies cos2 x = 1 − sin2 x, thus..
1 − 2 sin2 x
(work on right side) 2
= 1 − 2(1 − cos x) 2
= 2 cos x − 1
(the tweaked sub) (aglebra)
18. Practice Work on each side: Assume there is an interesting world in which x2 + y 2 = 1 for all values of x and y. In such world, Determine if the following is an identity, if so answer TRUE, if not answer FALSE. (x + y)2 = 2xy + 1 A. True B. False
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math hands Trigonometry 05 exercises Solution: work on the left side again.. (x + y)2
(left side)
2
(Expand)
2
x + 2xy + y 2
2
2xy + (x + y )
(Commute, associate, prep to sub)
2xy + (1)
(Commute, associate, prep to sub)
Therefore TRUE 19. Practice Work on each side: Determine if the following is an identity, prove your answer. (cos x + sin x)2 = 2 cos x sin x + 1 A. True B. False Solution: work on the left side again.. (cos x + sin x)2 2
(left side)
2
cos x + 2 cos x sin x + sin x 2
(Expand)
2
2 cos x sin x + (cos x + sin x)
(Commute, associate, prep to sub)
2 cos x sin x + (1)
(Commute, associate, prep to sub)
Therefore TRUE 20. Practice Work on each side: Determine if the following is an identity, prove your answer. cos4 x − sin4 x = 2 cos4 x − 2 cos2 x + 1 A. True B. False 21. Practice Work on each side: Determine if the following is an identity, prove your answer. (cos x + sin x)4 = −4 cos4 x + 4 cos2 x + 4 cos x sin x + 1 A. True B. False 22. Practice Work on each side: Determine if the following is an identity, prove your answer. cos x − sin x =
1 cos x + sin x
A. True B. False
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math hands Trigonometry 05 exercises 23. Practice Work on each side: Determine if the following is an identity, prove your answer. 1 1 2 cos x + = cos x + sin x cos x − sin x 1 − 2 sin2 x A. True B. False 24. Practice Work on each side: Determine if the following is an identity, prove your answer. 1 2 cos x 1 + = cos x + sin x cos x − sin x 2 cos2 x − 1 A. True B. False 25. Practice Work on each side: Determine if the following is an identity, prove your answer. 1 1 = 2 2 2 cos x − 1 1 − 2 sin x A. True B. False 26. Practice Work on each side: Determine if the following is an identity, prove your answer. cos x − sin x =
1 2 cos x + sin x
A. True B. False 27. Practice Work on each side: Determine if the following is an identity, prove your answer. 1 1 cos x − = cos x + sin x cos x − sin x 1 − 2 sin2 x A. True B. False 28. Practice Work on each side: Determine if the following is an identity, prove your answer. 1 2 cos x 1 + = cos x + sin x cos x − sin x 1 − 2 cos2 x A. True B. False 29. Practice Work on each side: Determine if the following is an identity, prove your answer. cos6 x + 3 cos4 x sin2 x + 3 cos2 x sin4 x + sin6 x + 2 = 3 A. True B. False 30. Practice Work on each side: Determine if the following is an identity, prove your answer. cos6 x + 3 cos4 x sin2 x + 3 cos2 x sin4 x + sin6 x − cos2 x = sin2 x A. True
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math hands Trigonometry 05 exercises B. False 31. Prove the following is an identity OR prove the following is not an identity cos6 θ = −2 + 3 cos2 θ + 3 sin4 θ − sin6 θ
32. Prove the following is an identity OR prove the following is not an identity 1 − cos4 θ = sin2 θ + sin2 θ cos2 θ
33. (**)Prove the identity sin(2x) = 2 sin x cos x
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