Question

1) sin x - √(3) cos x=0; 2) 2sin x + cos x =0; 3) sin^2x - 5 sin x cos x+4 cos^2 =0; 4) 3sin^2 - 2√(3) sin x cos x + cos^2x=0.

Answer 1

English Launches:

sin x - √(3) cos x = 0

Divide both sides by cos x (assuming cos x is not equal to 0):

sin x / cos x - √(3) = 0 / cos x

tan x = √(3)

x = π/3 + kπ, where k is an integer

2sin x + cos x = 0

Divide both sides by cos x (assuming cos x is not equal to 0):

2tan x + 1 = 0

tan x = -1/2

x = 7π/6 + kπ, where k is an integer

sin^2x - 5 sin x cos x+4 cos^2x = 0

This equation can be factored as:

(sin x - 4 cos x)(sin x - cos x) = 0

So either sin x - 4 cos x = 0 or sin x - cos x = 0.

If sin x - 4 cos x = 0:

Divide both sides by cos^2x (assuming cos x is not equal to 0):

tan x - 4 = 0

tan x = 4

x = arctan(4) + kπ, where k is an integer

If sin x - cos x = 0:

sin x = cos x

Divide both sides by cos^2x (assuming cos x is not equal to 0):

tan x = 1

x = π/4 + kπ, where k is an integer

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

This equation can be written as:

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

Since the square of a real number is always non-negative, the left-hand side of the equation is always greater than or equal to 0. Therefore, there are no real solutions for this equation.

Українська мова:

sin x - √(3) cos x = 0

Ми можемо розкласти ліву частину на добуток sin x та cos x:

sin x - √(3) cos x = 0

sin x / cos x = √(3)

tg x = √(3)

x = π/3 + πk, де k - ціле число.

2sin x + cos x = 0

Можемо поділити обидві частини на √(5) та використати формули для sin та cos кутів суми:

2sin x / √(5) + cos x / √(5) = 0

sin (x+y) = 0, де y = arccos (-2/√(5))

x = πk - y, де k - ціле число.

sin^2x - 5 sin x cos x + 4 cos^2x = 0

Ми можемо переписати це рівняння в такому вигляді:

(sin x - 4cos x)(sin x - cos x) = 0

Тоді ми маємо два випадки:

sin x - 4cos x = 0

tg x = 4

x = arctg 4 + πk, де k - ціле число.

або

sin x - cos x = 0

tg x = 1

x = π/4 + πk, де k - ціле число.

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

Можемо переписати це рівняння таким чином:

(√(3)sin x - cos x)(sin x - √(3)cos x) = 0

Тоді ми маємо два випадки:

√(3)sin x - cos x = 0

tg x = 1/√(3)

x = π/6 + πk, де k - ціле число.

або

sin x - √(3)cos x = 0

tg x = √(3)

x = π/3 + πk, де k - ціле число.

Answer 2

To solve each of these trigonometric equations, we can use various trigonometric identities and techniques. Let's solve each equation one by one:

1. sin(x) - √3 cos(x) = 0:

We can use the trigonometric identity sin(π/3) = √3/2 and cos(π/3) = 1/2 to simplify this equation:

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

Now, we need to find the values of x for which sin(x) = 2sin(π/3 + x). We can use the identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b):

sin(x) = 2(sin(π/3)cos(x) + cos(π/3)sin(x)) sin(x) = 2(√3/2 cos(x) + 1/2 sin(x))

Now, we can rewrite sin(x) in terms of cos(x):

sin(x) = √3 cos(x) + sin(x)

Now, move sin(x) to the left side:

√3 cos(x) = 0

To solve for x, divide both sides by √3:

cos(x) = 0

Now, we know that cos(x) = 0 when x is an odd multiple of π/2:

x = π/2 + kπ, where k is an integer.

2. 2sin(x) + cos(x) = 0:

This equation can be simplified by dividing both sides by cos(x):

2sin(x)/cos(x) + 1 = 0

Now, use the identity tan(x) = sin(x)/cos(x):

2tan(x) + 1 = 0

2tan(x) = -1

tan(x) = -1/2

To find the solutions for x, we can take the arctan of both sides:

x = arctan(-1/2) + kπ, where k is an integer.

3. sin^2(x) - 5sin(x)cos(x) + 4cos^2(x) = 0:

This equation can be factored:

(sin(x) - 4cos(x))(sin(x) - cos(x)) = 0

Now, we have two cases:

Case 1: sin(x) - 4cos(x) = 0

sin(x) = 4cos(x)

tan(x) = 4

x = arctan(4) + kπ, where k is an integer.

Case 2: sin(x) - cos(x) = 0

sin(x) = cos(x)

tan(x) = 1

x = arctan(1) + kπ, where k is an integer.

4. 3sin^2(x) - 2√3sin(x)cos(x) + cos^2(x) = 0:

This equation can be factored as well:

(√3sin(x) - cos(x))(3sin(x) - cos(x)) = 0

Now, we have two cases:

Case 1: √3sin(x) - cos(x) = 0

√3sin(x) = cos(x)

tan(x) = 1/√3

x = arctan(1/√3) + kπ, where k is an integer.

Case 2: 3sin(x) - cos(x) = 0

3sin(x) = cos(x)

tan(x) = 1/3

x = arctan(1/3) + kπ, where k is an integer.