quick question: is '[Cos(3x)]^2+[sin(3x)]^2=1' true?

[mammothrob]

[Cos(x)]^2+[sin(x)]^2=1

is this true?

[Cos(3x)]^2+[sin(3x)]^2=1

I tried some values for x on my calculator, and it was alright.

And will this work for any values for the arguments of the trig functions, even if they are composite functions... just as long as they are both the same for cos and sin?

thanx

 

[pka]

Re: quick question

[Cos(x)]^2+[sin(x)]^2=1, is this true?
[Cos(3x)]^2+[sin(3x)]^2=1

\(\displaystyle \L\cos ^2 \left( \theta \right) + \sin ^2 \left( \theta \right) = 1\) is true for any \(\displaystyle \theta\).
Even \(\displaystyle \theta = 3x\)

 

[soroban]

Re: quick question

Hello, mammothrob!

It's gratifying that you're curious enough to do some experimenting.

\(\displaystyle \cos^2x\,+\,\sin^2x\:=\:1\)

Is this true? \(\displaystyle \:\cos^2(3x)\,+\,\sin^2(3x)\:=\:1\)

I tried some values for x on my calculator, and it was alright.

Will this work for any values for the arguments of the trig functions? . . . . Yes!

The identity: \(\displaystyle \,\sin^2\theta\,+\,\cos^2\theta\:=\:1\,\) holds for any value of \(\displaystyle \theta\).

\(\displaystyle \theta\) could be \(\displaystyle 5x\) . . . or \(\displaystyle y^2\,+\,7\) . . . or \(\displaystyle e^z\) . . . or \(\displaystyle \ln\left(x^3\,+\,\frac{\pi}{4}\right)\) . . . etc.

 

[Denis]

What it really is is a right triangle with hypotenuse = 1, legs = cos(x) and sin(x)

Do a http://www.google.com search on "introduction to trigonometry"