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Math Help - Inverse Trig Functions

  1. #1
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    Inverse Trig Functions

    I'm not getting this stuff at all. Finding math help online is quite hard, so I hope you guys can help me.

    How do you solve problems like these ones:

    sin(arccos(3/5) + arccos(5/13))

    and

    cos(2arcsin(-2/3))

    How do you find the arccos of something like (3/5) without a calculator?!

    I have some more questions, but I'll post them later for sake of clarity.

    Thanks Everyone
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  2. #2
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    Hello Chicken Enchilada

    Welcome to Math Help Forum.

    You've come to the right place!
    Quote Originally Posted by Chicken Enchilada View Post
    I'm not getting this stuff at all. Finding math help online is quite hard, so I hope you guys can help me.

    How do you solve problems like these ones:

    sin(arccos(3/5) + arccos(5/13))

    and

    cos(2arcsin(-2/3))

    How do you find the arccos of something like (3/5) without a calculator?!

    I have some more questions, but I'll post them later for sake of clarity.

    Thanks Everyone
    To solve problems involving inverse trig functions, like \arccos(\tfrac35), let x (or some variable) equal the inverse function and then express it the other way round; that is, in terms of the original trig function. So here:

    Let x = \arccos(\tfrac35) and y = \arccos(\tfrac{5}{13}).

    Then \cos x = \tfrac35 and \cos y = \tfrac{5}{13}, and we want to find \sin(x+y)

    Now expand: \sin(x+y) = \sin x\cos y+ \cos x \sin y

    We know \cos x and \cos y. So all that remains is to find \sin x and \sin y, which we can do using Pythagoras' Theorem (or \cos^2x + \sin^2x = 1, which is essentially the same thing).

    So, \sin x = \tfrac45 (Can you see why?)

    and \cos x = ... ?

    So \sin(x+y) = ... ?

    Can you complete it now?

    (And for the second one, use \cos2x = 1 -2\sin^2x)

    Grandad
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  3. #3
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    Quote Originally Posted by Grandad View Post
    So, \sin x = \tfrac45 (Can you see why?)


    Quote Originally Posted by Grandad View Post
    \cos x = ... ?
    I assume you mean sin(y)? cos(x) = (3/5).

    I got siny = (12/13). Plugged that in.....

    (4/5)(5/13) + (12/13)(3/5) = (56/65)

    Viola!

    and for the second problem:

    cos(2x) Abstract the arcsin(-2/3).
    x = arcsin(-2/3)

    sin(x) = (-2/3) Put it in terms of sin(X).

    Plug it in and solve.
    cos(2x) = 1 - 2(-2/3)^2

    (9/9) - (8/9) = (1 / 9)

    Yes!

    This makes sense now. My Prof is one of those old man Profs that just goes up to the board and starts writing math equations. I can kind of see where he is coming from when he gave us stuff like sin(arcos(x)) = \sqrt (1-x^2). It looks similar to the identity.

    What are the least amount of theorems/identities that I need to know? I can permute them as needed (I think). Here are two so far.




    The other questions I have concern derivatives and integrals so I'll post that in the Calc section.

    Thanks a bunch Grandad!
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  4. #4
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    Hello, Chicken Enchilada1

    What are the least amount of theorems/identities that I need to know?
    I can permute them as needed (I think). . . . . Good for you!

    Here are two so far:

    . . [1]\;\;\sin^2\!\theta + \cos^2\!\theta \;=\;1

    . . [2]\;\;\cos2\theta \;=\;1-2\cos^2\!\theta

    The first one, [1], is very important.

    . . Its two variations are good to know: . \begin{array}{c}\tan^2\!\theta + 1 \;=\;\sec^2\!\theta \\ \cot^2\!\theta + 1 \;=\;\csc^2\!\theta \end{array}


    The second one, [2], has three variations: . \cos2\theta \;=\;\begin{Bmatrix}2\cos^2\!\theta - 1 \\ 1 - 2\sin^2\!\theta \\ \cos^2\!\theta - \sin^2\!\theta \end{Bmatrix}

    If you know one of them, you can derive the other two (by substituting [1]),
    . . but knowing all three saves a lot of time.


    There is a third one you should know: . [3]\;\;\sin2\theta \;=\;2\sin\theta\cos\theta



    Then there are these identities from basic trig:

    . . \begin{array}{c} \csc\theta \:=\:\dfrac{1}{\sin\theta} \\ \\[-3mm] \sec\theta \:=\:\dfrac{1}{\cos\theta} \\ \\[-3mm] \cot\theta \:=\:\dfrac{1}{\tan\theta} \end{array}\qquad\quad \begin{array}{c}\tan\theta \:=\:\dfrac{\sin\theta}{\cos\theta} \\ \\[-3mm] \cot\theta \:=\:\dfrac{\cos\theta}{\sin\theta} \end{array}

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  5. #5
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    Hello Chicken Enchilada
    Quote Originally Posted by Chicken Enchilada View Post



    I assume you mean sin(y)? cos(x) = (3/5).
    Yes of course I did. Sorry - but you've got it sorted in spite of me!

    What are the least amount of theorems/identities that I need to know? I can permute them as needed (I think). Here are two so far.




    The other questions I have concern derivatives and integrals so I'll post that in the Calc section.

    Thanks a bunch Grandad!
    That's a big question. I can't count the number of trig identities that I've used/learned over the years, but I'm afraid it probably runs to a few dozen. I couldn't say what the least number you'll need is - it depends very much on the level that you need to go to.

    Just to give you some idea, here's a Wikipedia page: List of trigonometric identities - Wikipedia, the free encyclopedia. But don't let this put you off - it's like learning any topic: just do what you need to at any given time. And if you're going to use this stuff regularly, you'll just pick it up bit by bit. As an art teacher I once knew said: practice, patience and perseverance!

    Grandad
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  6. #6
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    Great Chicken Enchilada, but who is Viola ?
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  7. #7
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    Thanks guys.
    Quote Originally Posted by HallsofIvy View Post
    Great Chicken Enchilada, but who is Viola ?

    La voici!
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