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Trigonometry Identity Review/Fun

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    i've already made a handful
    of videos that covers what
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    I'm going to cover, the
    trigonometric identities I'm
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    going to cover in this video.
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    The reason why I'm doing it is
    that I'm in need of review
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    myself because I was doing some
    calculus problems that required
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    me to know this, and I have
    better recording software now
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    so I thought two birds with one
    stone, let me rerecord a
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    video and kind of refresh
    things in my own mind.
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    So the trig identities that I'm
    going to assume that we know
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    because I've already made
    videos on them and they're a
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    little bit involved to remember
    or to prove, are that the sine
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    of a plus b is equal to the
    sine of a times the cosine
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    of b plus the sine of b
    times the cosine of a.
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    That's the first one, I assume,
    going into this video we know.
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    And then if we wanted to know
    the sine of-- well, I'll just
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    write it a little differently.
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    What if I wanted to figure out
    the sine of a plus-- I'll
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    write it this way-- minus c?
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    Which is the same thing
    as a minus c, right?
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    Well, we could just use this
    formula up here to say well,
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    that's equal to the sine of a
    times the cosine of minus c
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    plus the sine of minus c
    times the cosine of a.
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    And we know, and I guess this
    is another assumption that
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    we're going to have to have
    going into this video, that the
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    cosine of minus c is equal
    to just the cosine of c.
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    That the cosine is
    an even function.
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    And you could look at that by
    looking at the graph of the
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    cosine function, or even at
    the unit circle itself.
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    And that the sine is
    an odd function.
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    That the sine of minus
    c is actually equal
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    to minus sine of c.
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    So we can use both of that
    information to rewrite the
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    second line up here to say that
    the sine of a minus c is equal
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    to the sine of a times
    the cosine of c.
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    Because cosine of minus
    c is the same thing
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    as the cosine of c.
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    Times the cosine of c.
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    And then, minus the sine of c.
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    Instead of writing this,
    I could write this.
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    Minus the sine of c
    times the cosine of a.
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    So that we kind of pseudo
    proved this by knowing this
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    and this ahead of time.
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    Fair enough.
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    And I'm going to use all of
    these to kind of prove a bunch
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    of more trig identities
    that I'm going to need.
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    So the other trig identity is
    that the cosine of a plus b is
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    equal to the cosine of a-- you
    don't mix up the cosines and
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    the sines in this situation.
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    Cosine of a times
    the sine of b.
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    And this is minus--
    well, sorry.
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    I just said you don't mix it
    up and then I mixed them up.
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    Times the cosine of b minus
    sine of a times the sine of b.
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    Now, if you wanted to know what
    the cosine of a minus b is,
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    well, you use these
    same properties.
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    Cosine of minus b, that's still
    going to be cosine on b.
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    So that's going to be the
    cosine of a times the cosine--
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    cosine of minus b is the
    same thing as cosine of b.
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    But here you're going to have
    sine of minus b, which is the
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    same thing as the
    minus sine of b.
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    And that minus will cancel that
    out, so it'll be plus sine
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    of a times the sine of b.
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    So it's a little tricky.
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    When you have a plus sign
    here you get a minus there.
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    When you don't minus
    sign there, you get
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    a plus sign there.
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    But fair enough.
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    I don't want to dwell on that
    too much because we have many
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    more identities to show.
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    So what if I wanted an
    identity for let's
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    say, the cosine of 2a?
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    So the cosine of 2a.
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    Well that's just the same thing
    as the cosine of a plus a.
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    And then we could use this
    formula right up here.
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    If my second a is just my b,
    then this is just equal to
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    cosine of a times the cosine
    of a minus the sine of
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    a times the sine of a.
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    My b is also an a in this
    situation, which I could
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    rewrite as, this is equal to
    the cosine squared of a.
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    I just wrote cosine of a times
    itself twice or times itself.
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    Minus sine squared of a.
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    This is one I guess
    identity already.
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    Cosine of 2a is equal to the
    cosine squared of a minus
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    the sine squared of a.
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    Let me box off my identities
    that we're showing
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    in this video.
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    So I just showed you that one.
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    What if I'm not satisfied?
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    What if I just want it
    in terms of cosines?
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    Well, we could break out
    the unit circle definition
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    of our trig functions.
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    This is kind of the most
    fundamental identity.
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    The sine squared of a
    plus the cosine squared
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    of a is equal to 1.
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    Or you could write that--
    let me think of the
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    best way to do this.
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    You could write that the sine
    squared of a is equal to
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    1 minus the cosine
    sign squared of a.
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    And then we could take this
    and substitute it right here.
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    So we could rewrite this
    identity as being equal to the
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    cosine squared of a minus
    the sine squared of a.
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    But the sine squared of
    a is this right there.
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    So minus-- I'll do it
    in a different color.
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    Minus 1 minus cosine
    squared of a.
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    That's what I just substituted
    for the sine squared of a.
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    And so this is equal to the
    cosine squared of a minus 1
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    plus the cosine squared of a.
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    Which is equal to--
    we're just adding.
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    I'll just continue
    on the right.
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    We have 1 cosine squared of a
    plus another cosine squared
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    of a, so it's 2 cosine
    squared of a minus 1.
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    And all of that is
    equal to cosine of 2a.
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    Now what if I wanted to get
    an identity that gave me
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    what cosine squared of
    a is in terms of this?
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    Well we could just
    solve for that.
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    If we add 1 to both sides of
    this equation, actually,
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    let me write this.
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    This is one of our
    other identities.
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    But if we add 1 to both sides
    of that equation we get 2 times
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    the cosine squared of a is
    equal to cosine of 2a plus 1.
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    And if we divide both sides of
    this by 2 we get the cosine
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    squared of a is equal to 1/2--
    now we could rearrange these
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    just to do it-- times 1
    plus the cosine of 2a.
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    And we're done.
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    And we have another identity.
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    Cosine squared of a, sometimes
    it's called the power reduction
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    identity right there.
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    Now what if we wanted
    something in terms of
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    the sine squared of a?
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    Well then maybe we could go
    back up here and we know from
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    this identity that the sine
    squared of a is equal to 1
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    minus cosine squared of a.
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    Or we could have
    gone the other way.
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    We could have subtracted sine
    squared of a from both sides
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    and we could have gotten--
    let me go down there.
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    If I subtracted sine squared of
    a from both sides you could get
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    cosine squared of a is equal
    to 1 minus sine squared of a.
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    And then we could go back into
    this formula right up here and
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    we could write down-- and I'll
    do it in this blue color.
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    We could write down that the
    cosine of 2a is equal to--
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    instead of writing a cosine
    squared of a, I'll write this-
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    is equal to 1 minus sine
    squared of a minus
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    sine squared of a.
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    So my cosine of 2a is equal to?
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    Let's see.
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    I have a minus sine squared
    of a minus another
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    sine squared of a.
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    So I have 1 minus 2
    sine squared of a.
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    So here's another identity.
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    Another way to write
    my cosine of 2a.
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    We're discovering a lot of ways
    to write our cosine of 2a.
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    Now if we wanted to solve for
    sine squared of 2a we could
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    add it to both sides
    of the equation.
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    So let me do that and I'll
    just write it here for
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    the sake of saving space.
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    Let me scroll down
    a little bit.
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    So I'm going to go here.
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    If I just add 2 sine squared
    of a to both sides of this, I
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    get 2 sine squared of a plus
    cosine of 2a is equal to 1.
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    Subtract cosine of
    2a from both sides.
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    You get 2 sine squared of a is
    equal to 1 minus cosine of 2a.
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    Then you divide both sides of
    this by 2 and you get sine
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    squared of a is equal to 1/2
    times 1 minus cosine of 2a.
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    And we have our other discovery
    I guess we could call it.
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    Our finding.
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    And it's interesting.
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    It's always interesting
    to look at the symmetry.
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    Cosine squared-- they're
    identical except for you have a
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    plus 2a here for the cosine
    squared and you have a minus
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    cosine of 2a here for
    the sine squared.
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    So we've already found a
    lot of interesting things.
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    Let's see if we can do
    anything with the sine of 2a.
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    Let me pick a new color
    here that I haven't used.
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    Well, I've pretty much
    used all my colors.
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    So if I want to figure out the
    sine of 2a, this is equal
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    to the sine of a plus a.
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    Which is equal to the sine of a
    times the co-- well, I don't
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    want to make it that thick.
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    Times the cosine of a plus--
    and this cosine of a,
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    that's the second a.
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    Actually, you could
    view it that way.
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    Plus the sine-- I'm just
    using the sine of a plus b.
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    Plus the sine of the
    second a times the
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    cosine of the first a.
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    I just wrote the same thing
    twice, so this is just people
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    to 2 sine of a, cosine of a.
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    That was a little bit easier.
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    So sine of 2a is equal to that.
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    So that's another result.
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    I know I'm a little bit tired
    by playing with all of
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    these sine and cosines.
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    And I was able to get all the
    results that I needed for my
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    calculus problem, so hopefully
    this was a good review for
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    you because it was a
    good review for me.
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    You can write these
    things down.
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    You can memorize them if you
    want, but the really important
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    take away is to realize that
    you really can derive all of
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    these formulas really from
    these initial formulas
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    that we just had.
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    And even these, I also have
    proofs to show you how to get
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    these from just the basic
    definitions of your
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    trig functions.
Title:
Trigonometry Identity Review/Fun
Description:

Revisiting the proofs of some trigonometry identities.

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Video Language:
English
Duration:
11:07
brettle edited English subtitles for Trigonometry Identity Review/Fun Apr 18, 2011, 2:26 AM
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