Question

1) sin^2(3x)=cos^2(3x) 2) tg^3(x)+2tg^2(x)+3tgx=0

Answer 1

1) cos^2(3x) = sin^2(3x)
cos^2(3x) - sin^2(3x) = cos(6x) = 0
6x = π/2 + πk
x = π/12 + πk/6
2) tg^3(x) + 2tg^2(x) + 3tg(x) = 0
tg(x) * (tg^2 + 2tg(x) + 3) = 0

tg(x) = 0
x = πk

tg^2 + 2tg(x) + 3 = 0
(tg(x) + 1)^2 + 2 = 0
(tg(x) + 1)^2 = -2 —>Ø

Answer 2

To solve the given equations, let's work on each one separately:

1. sin^2(3x) = cos^2(3x)

Using the trigonometric identity: sin^2(x) + cos^2(x) = 1

We can rewrite the equation as:

sin^2(3x) + cos^2(3x) = 1

Now, apply the double angle formula for sine and cosine:

sin^2(2x + x) + cos^2(2x + x) = 1

Using the double angle formula:

sin^2(2x)cos^2(x) + cos^2(2x)sin^2(x) = 1

Since sin^2(x) + cos^2(x) = 1, we can replace sin^2(x) in the equation with (1 - cos^2(x)):

(1 - cos^2(2x))cos^2(x) + cos^2(2x)(1 - cos^2(x)) = 1

Now, expand and simplify:

cos^2(x) - cos^2(2x)cos^2(x) + cos^2(2x) - cos^2(2x)cos^2(x) = 1

Combine like terms:

cos^2(x) + cos^2(2x)(1 - cos^2(x)) = 1

Using the identity: cos^2(x) + sin^2(x) = 1

We can replace cos^2(x) with (1 - sin^2(x)):

(1 - sin^2(x)) + cos^2(2x)(1 - (1 - sin^2(x))) = 1

Now, simplify:

1 - sin^2(x) + cos^2(2x)sin^2(x) = 1

Subtract 1 from both sides:

-sin^2(x) + cos^2(2x)sin^2(x) = 0

Factor out sin^2(x):

sin^2(x)(cos^2(2x) - 1) = 0

Now, set each factor equal to zero:

sin^2(x) = 0 or cos^2(2x) - 1 = 0

sin(x) = 0 or cos^2(2x) = 1

The first equation sin(x) = 0 gives us the solutions: x = 0, x = π, x = 2π, x = 3π, and so on.

For the second equation, we have:

cos^2(2x) = 1

Using the identity: cos^2(x) = 1 - sin^2(x)

1 - sin^2(2x) = 1

sin^2(2x) = 0

Now, set sin^2(2x) = 0 equal to zero:

sin(2x) = 0

This gives us the solutions: 2x = 0, 2x = π, 2x = 2π, 2x = 3π, and so on.

Finally, divide each solution for 2x by 2 to find the values of x:

x = 0, x = π/2, x = π, x = 3π/2, and so on.

So the solutions to the equation sin^2(3x) = cos^2(3x) are: x = 0, x = π/2, x = π, x = 3π/2, and so on.

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2. tg^3(x) + 2tg^2(x) + 3tg(x) = 0

Since "tg" likely refers to the tangent function (tan(x)), let's use that notation:

tan^3(x) + 2tan^2(x) + 3tan(x) = 0

This is a cubic equation in terms of the tangent function. Unfortunately, there isn't a simple general formula to find the exact solutions for a cubic equation.

However, we can use numerical methods or graphing techniques to approximate the solutions. There might be specific values that satisfy this equation, but it's challenging to find them analytically. If you need specific approximations, you can use numerical methods or graphing calculators/software to find the solutions.