Math Is Fun Forum

rahulnayak89

Member

Registered: 2005-08-16

Posts: 2

Help me prove this.

Need help in proving this:

tan 9-tan 27 - tan 63 + tan81 = 4

Thanks in advance

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Help me prove this.

You don't really need to do anything to that, it's true already. The only thing that you can do to it is calculate the values that the tan functions give.

0.159-0.510-1.963+6.314=4, where all the numbers are rounded to 3 decimal places.

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Why did the vector cross the road?
It wanted to be normal.

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Help me prove this.

Some things can't be "proved" because they are already true based on the laws or definitions of mathematics.

We cannot prove that 1+1=2, because a basic law of mathematics says you CAN add, and in our number system "2" represents the amount that is "1" past "1" on the number line. Or something like that.

So, if you can get your calculator out and press the keys and the results match up, then it is simply true becasue most calculator programs are built using the rules and laws of mathematics (hmmm... that gives me an idea for a "Crazy Calculator!")

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

rahulnayak89

Member

Registered: 2005-08-16

Posts: 2

Re: Help me prove this.

I need to clarify.

You can not use a calculator.

You know the values of tan 0, tan 30, tan 45, tan 60 and tan 90.

With this knowledge, you will have to prove that the LHS is 4 using theorems of trigonometry.

I hope this clarifies.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,649

Re: Help me prove this.

hi

It's a trig identity to be proved.  Working on it but no luck yet.

I'll throw in two things that may be useful.

81 = 45 + 36
63 = 45 + 18
27 = 45  - 18
9  =  45 -  36

secondly

tan81 = 1/tan9
tan63 = 1 /tan 27

You'd think after five pages of trig manipulations I'd get there, but no.  I'll keep trying though!

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Bob

Administrator

Registered: 2010-06-20

Posts: 10,649

Re: Help me prove this.

hi

Ok, here we go:

Using tan(90 - x) = 1/tan x

using 1 + tan squared = sec squared

and then re-write using sec = 1/cos and also 1/tan = cos/sin .

Cancel terms and for the second step use sin 2x = 2 sinx cos x

This step took me a while.  It uses the factor formulas.

sin (P+Q) - sin (P-Q) = 2 cos P sin Q

Put P+Q = 54 and P-Q = 18 and find P = 36, Q = 18

Then you can cancel the sin 18 and also cos 36 cancels with the sin 54.

QED. 

Bob

Last edited by Bob (2011-10-28 21:05:05)

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob