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Math Help - Please Help with these problems

  1. #1
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    Please Help with these problems

    1) Determine the values of the trigonometric functions of the angle (smallest positive angle) if P is a point on the terminal side and is the coordinates of P are P( -1, -3).

    2)Find the diameter of a pulley which is driven at 360 r/min by a belt moving at 40 ft/s.

    3) State the quadrant in which the angle terminates and the signs of the sine, cosine, and tangent of the angle of 212 degrees.

    The teacher never went over this and i'm at a complete loss.
    Please help :[
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  2. #2
    MHF Contributor Reckoner's Avatar
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    Quote Originally Posted by Lollerpoppe View Post
    1) Determine the values of the trigonometric functions of the angle (smallest positive angle) if P is a point on the terminal side and is the coordinates of P are P( -1, -3).
    Look:



    You have a right triangle, with sides of length 3, 1, and \sqrt{10}. Surely you can evaluate the trigonometric functions here (if not, start paying attention!). Just remember that these lengths are "directed," so you need to take into account the negative signs. For example,

    Although for our angle \theta, \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac1{\sqrt{10}} =  \frac{\sqrt{10}}{10}, the angle \alpha made with the positive x-axis is in quadrant III, so the cosine is negative and we have \cos\alpha = -\frac{\sqrt{10}}{10}.

    You should be able to do the rest in similar fashion.

    Quote Originally Posted by Lollerpoppe View Post
    2)Find the diameter of a pulley which is driven at 360 r/min by a belt moving at 40 ft/s.
    If the pulley is rotating at 360\text{ rad/min} = 6\text{ rad/s}, then a point on the edge of the pulley will be traveling at 6r\text{ ft/s}, where r is the radius of the pulley in feet. So we have 6r = 40.

    Quote Originally Posted by Lollerpoppe View Post
    3) State the quadrant in which the angle terminates and the signs of the sine, cosine, and tangent of the angle of 212 degrees.
    These are the quadrants, along with the signs of the trigonometric functions in each:



    A mnemonic that is sometimes used for the signs is:

    All - All three (sin, cos, and tan) are positive in QI
    Students - Only sin is positive in QII
    Take - Only tan is positive in QIII
    Calculus - Only cos is positive in QIV

    Another way of looking at it is to use the unit circle:



    By drawing a perpendicular to the x-axis, you can see that every point on the unit circle is of the form (\cos\theta,\;\sin\theta), where \theta is the angle between the positive x-axis and the radial line containing the point. Thus you can easily see that the sine is positive wherever y is positive (quadrants I and II), and the cosine is positive wherever x is positive (quadrants I and IV).
    Attached Thumbnails Attached Thumbnails Please Help with these problems-trig3.png   Please Help with these problems-quadrants.png   Please Help with these problems-ucircle.png  
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  3. #3
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Reckoner View Post
    A mnemonic that is sometimes used for the signs is:

    All - All three (sin, cos, and tan) are positive in QI
    Students - Only sin is positive in QII
    Take - Only tan is positive in QIII
    Calculus - Only cos is positive in QIV
    Adding/Changing something to this:

    A mnemonic that is sometimes used for the signs is:

    All - All six (sin, cos, tan, sec, csc, and cot) are positive in QI
    Students - Only sin and csc is positive in QII
    Take - Only tan and cot is positive in QIII
    Calculus - Only cos and sec is positive in QIV
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  4. #4
    MHF Contributor Reckoner's Avatar
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    Quote Originally Posted by Chris L T521 View Post
    Adding/Changing something to this:

    A mnemonic that is sometimes used for the signs is:

    All - All six (sin, cos, tan, sec, csc, and cot) are positive in QI
    Students - Only sin and csc is positive in QII
    Take - Only tan and cot is positive in QIII
    Calculus - Only cos and sec is positive in QIV
    Thanks. I'm not sure why I didn't mention the others, but being reciprocals, the mnemonic certainly holds for them as well.
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