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Simplifying trigonometric ratios

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  1. Offline

    ReputationRep:
    I'm doing a few exercises where you need to simplify trigonometric ratios of either 30, 45 to 60 degrees

    Here are the examples it gave.

    cos(405)
    cos(405) = cos(360+45)
    =cos(45)

    tan(120)
    tan(120) = tan(180-60)
    =tan(-60)
    =-tan(60)

    sin(300)
    sin(300)=sin(360-60)
    =sin(-60)
    =-sin(60)

    so it seems that the rules are
    sin(-x) = -sin(x), cos(-x) = -cos(x), tan(-x) = -tan(x)
    sin(360+x) = sin(-360+x) = sin(x)
    cos(360+x) = cos(-360+x) = cos(x)
    tan(180+x) = tan(-180+x) = tan(x)

    Let's say you have
    cos(-210)
    adding 360 won't help you in any way as you will just be left with cos(150) and the examples never bothered to show how to handle this situation. but I think you can just add or remove 180 and flip the sign

    cos(-210)= -cos(-210 + 180) = -cos(-30)
    = --cos(30) = cos(30), but this is wrong. So "cos(-x) = -cos(x)" does not always work. Are there any consistent rules for how to simplify these? The ones the example is showing don't even work.
  2. Offline

    ReputationRep:
    (Original post by Bobby132)
    I'm doing a few exercises where you need to simplify trigonometric ratios of either 30, 45 to 60 degrees

    Here are the examples it gave.

    cos(405)
    cos(405) = cos(360+45)
    =cos(45)

    tan(120)
    tan(120) = tan(180-60)
    =tan(-60)
    =-tan(60)

    sin(300)
    sin(300)=sin(360-60)
    =sin(-60)
    =-sin(60)

    so it seems that the rules are
    sin(-x) = -sin(x), cos(-x) = -cos(x), tan(-x) = -tan(x)
    sin(360+x) = sin(-360+x) = sin(x)
    cos(360+x) = cos(-360+x) = cos(x)
    tan(180+x) = tan(-180+x) = tan(x)

    Let's say you have
    cos(-210)
    adding 360 won't help you in any way as you will just be left with cos(150) and the examples never bothered to show how to handle this situation. but I think you can just add or remove 180 and flip the sign

    cos(-210)= -cos(-210 + 180) = -cos(-30)
    = --cos(30) = cos(30), but this is wrong. So "cos(-x) = -cos(x)" does not always work. Are there any consistent rules for how to simplify these? The ones the example is showing don't even work.
    cos(-x)=cos(x)
  3. Offline

    ReputationRep:
    (Original post by Bobby132)
    I'm doing a few exercises where you need to simplify trigonometric ratios of either 30, 45 to 60 degrees

    Here are the examples it gave.

    cos(405)
    cos(405) = cos(360+45)
    =cos(45)

    tan(120)
    tan(120) = tan(180-60)
    =tan(-60)
    =-tan(60)

    sin(300)
    sin(300)=sin(360-60)
    =sin(-60)
    =-sin(60)

    so it seems that the rules are
    sin(-x) = -sin(x), cos(-x) = -cos(x), tan(-x) = -tan(x)
    sin(360+x) = sin(-360+x) = sin(x)
    cos(360+x) = cos(-360+x) = cos(x)
    tan(180+x) = tan(-180+x) = tan(x)

    Let's say you have
    cos(-210)
    adding 360 won't help you in any way as you will just be left with cos(150) and the examples never bothered to show how to handle this situation. but I think you can just add or remove 180 and flip the sign

    cos(-210)= -cos(-210 + 180) = -cos(-30)
    = --cos(30) = cos(30), but this is wrong. So "cos(-x) = -cos(x)" does not always work. Are there any consistent rules for how to simplify these? The ones the example is showing don't even work.
     sin(-x)= -sin(x) \\ tan(-x) = -tan(x) \\ cos(-x) \not= -cos(x) \\ cos(-x) = cos(x)
  4. Offline

    ReputationRep:
    Rather than trying to learn rules that you are not even sure of it would be better to use the graphs or the CAST diagram

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