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# Simplifying trigonometric ratios

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1. 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. (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. (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.
4. 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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