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How do you express trigonometric identites in simplest form, e.g. $frac2sec(x)tan^2(x)$?
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Are there strict rules defining what the "simplest" form of a trigonometric identity is? For example, $dfrac2sec(x)tan^2(x)$ can also be expressed as $dfrac2cos(x)sin^2(x)$ [ignoring restrictions on $x$].
Which one, if either of these forms, is "simpler," and why?
trigonometry
edited Jul 31 at 17:59
amWhy
189k25219431
asked Jul 31 at 17:46
Matthew Sylvester
335
add a comment |Â
up vote
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down vote
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Are there strict rules defining what the "simplest" form of a trigonometric identity is? For example, $dfrac2sec(x)tan^2(x)$ can also be expressed as $dfrac2cos(x)sin^2(x)$ [ignoring restrictions on $x$].
Which one, if either of these forms, is "simpler," and why?
trigonometry
edited Jul 31 at 17:59
amWhy
189k25219431
asked Jul 31 at 17:46
Matthew Sylvester
335
1
The rule of thumb employed by computer algebra software (such as Mathematica) seems to be 1) the smallest number of trigonometric functions, then 2) the smallest number of arguments (e.g., $theta$s throughout is better than $theta$ and $2 theta$ and $theta/2$...
– David G. Stork
Jul 31 at 17:49
@DavidG.Stork Does rule 2 apply to the functions themselves too? If not, doesn't that mean that sin(x) and 1/csc(x) have the same "degree" of simplification?
– Matthew Sylvester
Jul 31 at 18:02
1
Don't know for sure.
– David G. Stork
Jul 31 at 18:03
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Are there strict rules defining what the "simplest" form of a trigonometric identity is? For example, $dfrac2sec(x)tan^2(x)$ can also be expressed as $dfrac2cos(x)sin^2(x)$ [ignoring restrictions on $x$].
Which one, if either of these forms, is "simpler," and why?
trigonometry
edited Jul 31 at 17:59
amWhy
189k25219431
asked Jul 31 at 17:46
Matthew Sylvester
335
Are there strict rules defining what the "simplest" form of a trigonometric identity is? For example, $dfrac2sec(x)tan^2(x)$ can also be expressed as $dfrac2cos(x)sin^2(x)$ [ignoring restrictions on $x$].
Which one, if either of these forms, is "simpler," and why?
trigonometry
edited Jul 31 at 17:59
amWhy
189k25219431
asked Jul 31 at 17:46
Matthew Sylvester
335
edited Jul 31 at 17:59
amWhy
189k25219431
edited Jul 31 at 17:59
amWhy
189k25219431
edited Jul 31 at 17:59
amWhy
189k25219431
189k25219431
asked Jul 31 at 17:46
Matthew Sylvester
335
asked Jul 31 at 17:46
Matthew Sylvester
335
asked Jul 31 at 17:46
Matthew Sylvester
335
335
1
The rule of thumb employed by computer algebra software (such as Mathematica) seems to be 1) the smallest number of trigonometric functions, then 2) the smallest number of arguments (e.g., $theta$s throughout is better than $theta$ and $2 theta$ and $theta/2$...
– David G. Stork
Jul 31 at 17:49
@DavidG.Stork Does rule 2 apply to the functions themselves too? If not, doesn't that mean that sin(x) and 1/csc(x) have the same "degree" of simplification?
– Matthew Sylvester
Jul 31 at 18:02
1
Don't know for sure.
– David G. Stork
Jul 31 at 18:03
add a comment |Â
1
The rule of thumb employed by computer algebra software (such as Mathematica) seems to be 1) the smallest number of trigonometric functions, then 2) the smallest number of arguments (e.g., $theta$s throughout is better than $theta$ and $2 theta$ and $theta/2$...
– David G. Stork
Jul 31 at 17:49
@DavidG.Stork Does rule 2 apply to the functions themselves too? If not, doesn't that mean that sin(x) and 1/csc(x) have the same "degree" of simplification?
– Matthew Sylvester
Jul 31 at 18:02
1
Don't know for sure.
– David G. Stork
Jul 31 at 18:03
1
1
The rule of thumb employed by computer algebra software (such as Mathematica) seems to be 1) the smallest number of trigonometric functions, then 2) the smallest number of arguments (e.g., $theta$s throughout is better than $theta$ and $2 theta$ and $theta/2$...
– David G. Stork
Jul 31 at 17:49
The rule of thumb employed by computer algebra software (such as Mathematica) seems to be 1) the smallest number of trigonometric functions, then 2) the smallest number of arguments (e.g., $theta$s throughout is better than $theta$ and $2 theta$ and $theta/2$...
– David G. Stork
Jul 31 at 17:49
@DavidG.Stork Does rule 2 apply to the functions themselves too? If not, doesn't that mean that sin(x) and 1/csc(x) have the same "degree" of simplification?
– Matthew Sylvester
Jul 31 at 18:02
@DavidG.Stork Does rule 2 apply to the functions themselves too? If not, doesn't that mean that sin(x) and 1/csc(x) have the same "degree" of simplification?
– Matthew Sylvester
Jul 31 at 18:02
1
1
Don't know for sure.
– David G. Stork
Jul 31 at 18:03
Don't know for sure.
– David G. Stork
Jul 31 at 18:03
add a comment |Â
1 Answer
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I would replace all of the trig forms with the letters A, B, and C (e.g. $sin(x) = B/C$ where B is the 'opposite' and C is the hypotenuse) because they represent the same triangle sides if all of the angles here are 'x'. Then, you can do the simplification using algebra and return to the trig form(s) when you are done. Surely, an algebraic collection and/or cancelation of terms would make the simplest form easier to see. Good luck.
answered Jul 31 at 18:19
poetasis
17513
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
I would replace all of the trig forms with the letters A, B, and C (e.g. $sin(x) = B/C$ where B is the 'opposite' and C is the hypotenuse) because they represent the same triangle sides if all of the angles here are 'x'. Then, you can do the simplification using algebra and return to the trig form(s) when you are done. Surely, an algebraic collection and/or cancelation of terms would make the simplest form easier to see. Good luck.
answered Jul 31 at 18:19
poetasis
17513
add a comment |Â
up vote
0
down vote
I would replace all of the trig forms with the letters A, B, and C (e.g. $sin(x) = B/C$ where B is the 'opposite' and C is the hypotenuse) because they represent the same triangle sides if all of the angles here are 'x'. Then, you can do the simplification using algebra and return to the trig form(s) when you are done. Surely, an algebraic collection and/or cancelation of terms would make the simplest form easier to see. Good luck.
answered Jul 31 at 18:19
poetasis
17513
add a comment |Â
up vote
0
down vote
up vote
0
down vote
I would replace all of the trig forms with the letters A, B, and C (e.g. $sin(x) = B/C$ where B is the 'opposite' and C is the hypotenuse) because they represent the same triangle sides if all of the angles here are 'x'. Then, you can do the simplification using algebra and return to the trig form(s) when you are done. Surely, an algebraic collection and/or cancelation of terms would make the simplest form easier to see. Good luck.
answered Jul 31 at 18:19
poetasis
17513
I would replace all of the trig forms with the letters A, B, and C (e.g. $sin(x) = B/C$ where B is the 'opposite' and C is the hypotenuse) because they represent the same triangle sides if all of the angles here are 'x'. Then, you can do the simplification using algebra and return to the trig form(s) when you are done. Surely, an algebraic collection and/or cancelation of terms would make the simplest form easier to see. Good luck.
answered Jul 31 at 18:19
poetasis
17513
answered Jul 31 at 18:19
poetasis
17513
answered Jul 31 at 18:19
poetasis
17513
answered Jul 31 at 18:19
poetasis
17513
17513
add a comment |Â
add a comment |Â
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1
The rule of thumb employed by computer algebra software (such as Mathematica) seems to be 1) the smallest number of trigonometric functions, then 2) the smallest number of arguments (e.g., $theta$s throughout is better than $theta$ and $2 theta$ and $theta/2$...
– David G. Stork
Jul 31 at 17:49
@DavidG.Stork Does rule 2 apply to the functions themselves too? If not, doesn't that mean that sin(x) and 1/csc(x) have the same "degree" of simplification?
– Matthew Sylvester
Jul 31 at 18:02
1
Don't know for sure.
– David G. Stork
Jul 31 at 18:03