S2W3 FC - Additive/subtractive identities - Gwyneth Lenfest | Library

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An identity is true no matter what value is chosen for its variable. so lets do some true-false questions: each of these is either an identity or not an identity. categorize them.

identity
not an identity

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which word needs to be changed?

When we proved that tan x= sin x/cos x we used similar triangles. A related proof is used to prove that secant is the reciprocal of cosine, but this time the secant is the leg of the same triangle used in the tan proof.

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Does the way these two pictures show the geometric forms of the six functions reconcile you a little to the fact that csc is the reciprocal of sin, not cosine?

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To put it all together

do you have any questions?

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Review: the co in cosine, cotangent and cosecant stands for complementary, which refers to the relationship to each other when their __________. The sine function is an __________ function, because the graph of one side of the y axis is __________. Another list of trig identities to know are the cofunction identities - 
Cos x= Sin __________
__________ = tan (90°-x)
__________=__________
__________=__________
__________=__________

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Draggable item		Corresponding Item

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Can you solve the following equation?

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Prove the following identity: it easiest to just use a bunch of equation tool doohickies.

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This is another section. I suggest you practice some of these in your modules and homework, and come back for the rest after you have let your mind brain absorb stuff.

Ok, our first set of identities was just relating all the trig functions together. Which is the reciprocal of what, what is the quotient of who, etc.

Then with odd/even function and co-functions, we had "what happens when we change the angle inside in specific ways, those ways being negating and phase shifting."
this next one is more general - what happens when we have two different angles being added together?
are there particular parts of this previous section you need more practice with?

Trim End | 08:26

Questions 12 & 13 | 00:44

Question 14 | 03:58

00:00/00:00

Questions 12 & 13

00:44

Question 14

03:58

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Pause here and remember what I said about never subtracting. Replace b with negative b. Which identity will you need to use this equation to figure out sin (a-b)

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Use the identities you picked before, change up sin(a+b) to sin(a+-b), and write down what you think it will be.

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See if you can work out cos(x+y) on this version of the same proof.

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OK, you should have gotten \cos(x+y)=\cos x \cos y-\sin x \sin y. Now use the even/odd identity to show what cos(x-y) should be.

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here I want you to rewrite sin(a+b) where b=a. What do you get?

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One more time, look at that equation, use the same identities you used before, and translate cos(a+-b)

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Prove the following identity.

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It is easy to put \sin(a+a) into the addition formula to find the formula for \sin 2a. Here are the equations for \sin a/2

do you want to go through the proof for these, or no?

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prove one of the cofunction identities using the angle addition identities?

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Last set of identities. These are just formulas you can use when they come up, they aren't ones you usually need to remember exist when being asked to prove an identity.
any questions?

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Using the above list and only using the unit circle, give an exact answer to
First use the two angles above to find u and v.

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how are you feeling?

what qualifies as an identity

Draggable item		Corresponding Item
csc x		cos (90°-x)
cot x		cot(90°-x)
cos(x)		csc (90°-x)
		1/sin x
sec x		cos x/sin x
tan x		1
sin x

		cos(-x)

S2W3 FC - Additive/subtractive identities

S2W3 FC - Additive/subtractive identities

An identity is true no matter what value is chosen for its variable. so lets do some true-false questions: each of these is either an identity or not an identity. categorize them.

which word needs to be changed?

Does the way these two pictures show the geometric forms of the six functions reconcile you a little to the fact that csc is the reciprocal of sin, not cosine?

To put it all togetherdo you have any questions?

Can you solve the following equation?

Prove the following identity: it easiest to just use a bunch of equation tool doohickies.

Pause here and remember what I said about never subtracting. Replace b with negative b. Which identity will you need to use this equation to figure out sin (a-b)

Use the identities you picked before, change up sin(a+b) to sin(a+-b), and write down what you think it will be.

See if you can work out cos(x+y) on this version of the same proof.

OK, you should have gotten \cos(x+y)=\cos x \cos y-\sin x \sin y. Now use the even/odd identity to show what cos(x-y) should be.

here I want you to rewrite sin(a+b) where b=a. What do you get?

One more time, look at that equation, use the same identities you used before, and translate cos(a+-b)

Prove the following identity.

It is easy to put \sin(a+a) into the addition formula to find the formula for \sin 2a. Here are the equations for \sin a/2do you want to go through the proof for these, or no?

prove one of the cofunction identities using the angle addition identities?

Last set of identities. These are just formulas you can use when they come up, they aren't ones you usually need to remember exist when being asked to prove an identity. any questions?

Using the above list and only using the unit circle, give an exact answer to First use the two angles above to find u and v.

how are you feeling?

Review questions: Categorize each parent function by dragging them to the different lables

An identity is true no matter what value is chosen for its variable. so lets do some true-false questions: each of these is either an identity or not an identity. categorize them.

which word needs to be changed?

what should it be changed to?

Does the way these two pictures show the geometric forms of the six functions reconcile you a little to the fact that csc is the reciprocal of sin, not cosine?

To put it all togetherdo you have any questions?

So far we have a set of basic trig identities, pythagorean identities and the cofunction identities, and odd/even identities. See if you can match identities,

Can you solve the following equation?

Prove the following identity: it easiest to just use a bunch of equation tool doohickies.

Pause here and remember what I said about never subtracting. Replace b with negative b. Which identity will you need to use this equation to figure out sin (a-b)

Use the identities you picked before, change up sin(a+b) to sin(a+-b), and write down what you think it will be.

See if you can work out cos(x+y) on this version of the same proof.

OK, you should have gotten \cos(x+y)=\cos x \cos y-\sin x \sin y. Now use the even/odd identity to show what cos(x-y) should be.

here I want you to rewrite sin(a+b) where b=a. What do you get?

One more time, look at that equation, use the same identities you used before, and translate cos(a+-b)

Prove the following identity.

It is easy to put \sin(a+a) into the addition formula to find the formula for \sin 2a. Here are the equations for \sin a/2do you want to go through the proof for these, or no?

prove one of the cofunction identities using the angle addition identities?

Last set of identities. These are just formulas you can use when they come up, they aren't ones you usually need to remember exist when being asked to prove an identity. any questions?

Using the above list and only using the unit circle, give an exact answer to First use the two angles above to find u and v.

how are you feeling?

what should it be changed to?

ok, lots of geometry. Which of the following is the same as angle y because of alternate interior angles

Which of the following is equal to (90°-y) Mark all that apply.

Since the hypotenuse of the red triangle is 1, we can assume that is the radius of a unit circle. which means,

To put it all together

do you have any questions?

It is easy to put \sin(a+a) into the addition formula to find the formula for \sin 2a. Here are the equations for \sin a/2

do you want to go through the proof for these, or no?

Last set of identities. These are just formulas you can use when they come up, they aren't ones you usually need to remember exist when being asked to prove an identity.
any questions?

Using the above list and only using the unit circle, give an exact answer to
First use the two angles above to find u and v.

To put it all together

do you have any questions?

It is easy to put \sin(a+a) into the addition formula to find the formula for \sin 2a. Here are the equations for \sin a/2

do you want to go through the proof for these, or no?

Last set of identities. These are just formulas you can use when they come up, they aren't ones you usually need to remember exist when being asked to prove an identity.
any questions?

Using the above list and only using the unit circle, give an exact answer to
First use the two angles above to find u and v.