Question

(1+√2cos(x+П/4)(tgx-3)=0

Answer 1

Если произведение равно 0, то любой из множителей может быть равен 0
1) 1 + √2cos(x + pi/4) = 0
cos(x + pi/4) = -1/√2
x1 + pi/4 = pi + pi/4 + 2pi*n
x1 = pi + 2pi*n
x2 = pi - pi/4 + 2pi*n
x2 = pi - pi/2 + 2pi*n = pi/2 + 2pi*n

2) tg x - 3 = 0
tg x = 3
x = arctg(3) + pi*k

Answer 2

To solve the equation (1 + √2cos(x+π/4)(tan(x)-3) = 0), we can follow these steps:

Step 1: Simplify the equation. Step 2: Solve for x using trigonometric identities and algebraic manipulations.

Let's go through these steps one by one:

Step 1: Simplify the equation. Using the trigonometric identity cos(x + π/4) = cos(x)cos(π/4) - sin(x)sin(π/4), we can simplify the equation as follows:

1 + √2cos(x+π/4)(tan(x)-3) = 0

1 + √2(cos(x)cos(π/4) - sin(x)sin(π/4))(tan(x)-3) = 0

1 + √2(cos(x)(√2/2) - sin(x)(√2/2))(tan(x)-3) = 0

1 + √2((√2/2)cos(x) - (√2/2)sin(x))(tan(x)-3) = 0

1 + (√2/2)(√2cos(x) - √2sin(x))(tan(x)-3) = 0

1 + (√2/2)(2cos(x) - 2sin(x))(tan(x)-3) = 0

1 + (√2)(cos(x) - sin(x))(tan(x)-3) = 0

Step 2: Solve for x using trigonometric identities and algebraic manipulations. Let's distribute the terms:

1 + (√2)(cos(x) - sin(x))(tan(x)-3) = 0

1 + (√2)(costan(x) - sintan(x) - 3cos(x) + 3sin(x)) = 0

Now, let's group the terms:

1 + (√2)(costan(x) - 3cos(x) - sintan(x) + 3sin(x)) = 0

Next, let's factor out common terms:

1 + (√2)(cos(x)(tan(x) - 3) - sin(x)(tan(x) - 3)) = 0

Now, we have (tan(x) - 3) as a common factor:

(tan(x) - 3)(1 + (√2)(cos(x) - sin(x))) = 0

Now we have two factors: (tan(x) - 3) = 0 and (1 + (√2)(cos(x) - sin(x))) = 0.

Solving (tan(x) - 3) = 0: tan(x) = 3

Using the unit circle or a calculator, we can find the values of x that satisfy this equation.

Solving (1 + (√2)(cos(x) - sin(x))) = 0: 1 + (√2)(cos(x) - sin(x)) = 0

√2(cos(x) - sin(x)) = -1

(cos(x) - sin(x)) = -1/√2

(cos(x) - sin(x)) = -√2/2

Let's define a new angle θ such that cos(θ) = 1/√2 and sin(θ) = 1/√2:

cos(x) - sin(x) = cos(θ) - sin(θ)

Using the identity cos(α - β) = cos(α)cos(β) + sin(α)sin(β), we can rewrite the equation as:

cos(x) - sin(x)