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Finding other Trig Values from one

  1. Aug 25, 2013 #1
    The problem gave that:

    theta= arcsin(-5/8)

    It wants us to find the other five trig function values.

    In order to do that, I turned the original function to
    sin(theta)= - 5/8

    From there I was able to find csc(theta) to be -8/5.

    Using the pythagorian theorem I found the other side of the triangle that can be formed to be sqrt(39)

    But because sin was negative from the original function, how are you supposed to figure out the quadrant the function lies in, because sin is negative in the 3rd and 4th quadrant.

    I understand how to find the other values, I just don't which quadrant it lies in because it would effect the positive and negative signs.
     
  2. jcsd
  3. Aug 25, 2013 #2

    ehild

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    theta= arcsin(-5/8) is not equivalent with sin(theta)= - 5/8. The range of the arcsin function is [-pi/2, pi/2]. So you can decide about the quadrant the angle lies in.
    Inverse trigonometric functions - Wikipedia, the free encyclopedia

    ehild
     
  4. Aug 25, 2013 #3
    Those two are equivalent because if you do sin45 it equals 1/(sqrt2) and if you do arcsin(1/sqrt2) it equals 45

    They are inverse functions.
     
  5. Aug 25, 2013 #4

    ehild

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    sinx has no inverse in its whole domain for -∞ to ∞, as it is not a monotonous function. It can be inverted between -pi/2 and pi/2 when it gives the principal values of arcsin function.

    ehild
     
  6. Aug 25, 2013 #5
    I do not have any idea what you are saying, but from what you are saying, what would the quadrant be? You are talking about a range from -pi/2 to pi/2 but I don't understand what you mean. We are not supposed to be using calculators so I don't know. Also, we need to find answers such as a side over another side, like the answer to costheta, or tantheta, which would require a conversion from arcsin to sin.
     
  7. Aug 25, 2013 #6

    ehild

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    Can you tell me what the quadrants are? And what are the signs of cosine and sine in them?

    ehild
     
  8. Aug 25, 2013 #7
    Quadrant 1: sin and cos are both postive

    Quadrant 2: sin is only positive, cos negative

    Quadrant 3: sin and cos are negative

    Quadrant 4: cos is only positive, sin is negative
     
  9. Aug 25, 2013 #8

    ehild

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    What angles belong to the individual quadrants? For example 300° belongs to which quadrant?

    ehild
     
  10. Aug 25, 2013 #9
    What do you mean by angles? Also, why does it matter?

    There are multiple angles per quadrant from 30 degrees to 60 degrees to 45 degrees and so on for each quadrant. Also, the -5/8 doesn't have a special angle like 30, 60, 90 triangles or the 45, 45, 90 triangles
     
  11. Aug 25, 2013 #10

    ehild

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    Well, f(x) is a trigonometric function. You speak abut quadrants defined by the sign of sine and cosines. What is the interval for x in the first quadrant, second, third, and so on?

    ehild
     
  12. Aug 25, 2013 #11
    I think we are going about this the wrong way. I know for a fact that it is not going to be in the first or second quadrant because the sin was negative. But I don't know weather it is in quadrant 3 or 4 because they both can have negative sines. It might have to do with ranges as you were talking about before. I am thinking quadrant 4 but my gut is saying quadrant 3 and I just want to know how to be able to tell.
     
  13. Aug 25, 2013 #12

    ehild

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    theta =arssin(x) returns theta values between -pi/2 and pi/2. Only a monotonous function has inverse. sin(theta) is monotonous in the range -pi/2≤theta ≤pi/2. -pi/2 is the same as 3pi/2, so the domain includes the fourth and first quadrants. According to the sign of the sine, theta can belong to the third or fourth quadrant, but theta in the third quadrant can not be the value of arcsin(-5/8). That means, theta is in the fourth quadrant, where the cosine is positive.


    http://www.marlenesite.com/math/trigonometry/images/quadrant.gif

    ehild
     
  14. Aug 25, 2013 #13
    Thank you, that makes sense but what do you mean by monotonous functions? I remember learning about things such as sin and tan only go from pi/2 to 3pi/2 and cos is the other way, but I don't really remember.
     
  15. Aug 25, 2013 #14

    ehild

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    You can invert a function only in the domain where it is monotonous. Otherwise f(x) can be the same at several different x-es, say f(x1)=f(x2). The inverse function returns the x value where f is given. There are two of them, but a function can have only one value.

    So you were taught that the range of arcsin is [pi/2; 3pi/2]? It is all right, sin(x) is monotonous also in that range, although [-pi/2, pi/2] is the usual range. Your calculator will return a negative value for the arcsin of a negative number, and taking the cosine of the result, it will be positive. Try.
    [pi/2; 3pi/2] includes the second and third quadrants, so you have to choose the cosine accordingly. Check your notes. Maybe, you need to give both the negative and positive cosines as answer.


    ehild
     
    Last edited: Aug 25, 2013
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