# Solve 3cos^2(3x)+3sin^2(3x)=3: Trig Identities

• mathguyz
In summary, solving 3cos^2(3x)+3sin^2(3x)=3 using trigonometric identities allows for simplification of the expression and finding the values of x that satisfy the equation. Common identities used include the Pythagorean identity, double angle formulas, and half-angle formulas. However, the equation can also be solved without using identities by factoring and using basic algebra. The possible solutions are all real numbers. This type of equation has various applications in fields such as physics, engineering, and astronomy.
mathguyz
Everyone knows the obvious trig identities like sin^2 + cos^2 =1, cosx=1+ sin^2, and tanx =sin/cos. I ran across an old identity the other day: 3cos^2(3x)+3sin^2(3x)=3. Can anyone here figure out why and how? I tried it and couldn't figure it out.

Look at your first "obvious trig identity"...

3cos^2(3x)+3sin^2(3x)=3[cos^2(3x)+sin^2(3x)] , now substitude 3x=z and using sin^2(z) + cos^2(z) =1 you have it!

NB: I´m not sure this is correct, whatever it is meant to be: cosx=1+ sin^2

## What is the purpose of solving 3cos^2(3x)+3sin^2(3x)=3 using trigonometric identities?

The purpose of solving this equation using trigonometric identities is to simplify the expression and find the value(s) of x that satisfy the equation. This can be useful in solving other equations and applications in fields such as physics, engineering, and astronomy.

## What are some common trigonometric identities used to solve 3cos^2(3x)+3sin^2(3x)=3?

Some common trigonometric identities used to solve this equation are the Pythagorean identity (sin^2(x) + cos^2(x) = 1), double angle formulas (cos(2x) = cos^2(x) - sin^2(x) and sin(2x) = 2sin(x)cos(x)), and the half-angle formulas (cos^2(x/2) = (1+cos(x))/2 and sin^2(x/2) = (1-cos(x))/2).

## Can 3cos^2(3x)+3sin^2(3x)=3 be solved without using trigonometric identities?

Yes, this equation can be solved without using trigonometric identities by factoring out a 3 and using the identity cos^2(x) + sin^2(x) = 1. The resulting equation, 3(cos^2(3x) + sin^2(3x)) = 3, simplifies to cos^2(3x) + sin^2(3x) = 1, which is the Pythagorean identity. From there, the equation can be solved using basic algebra.

## What are the possible solutions to 3cos^2(3x)+3sin^2(3x)=3?

Since this equation simplifies to cos^2(3x) + sin^2(3x) = 1, the possible solutions are values of x that satisfy this identity. This includes all real numbers, as any angle input into the functions cosine and sine will result in a value between -1 and 1, making the sum of their squares equal to 1.

## How can solving 3cos^2(3x)+3sin^2(3x)=3 be applied in real-world scenarios?

This type of equation can be applied in real-world scenarios such as calculating the amplitude and period of a sound wave or analyzing the motion of a pendulum. It can also be used in engineering to determine the strength and stability of structures and in astronomy to calculate the position and movement of celestial objects.

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