Question

Б) sin 2x = 1/2 В) sin x/2 = 1/2 2. А) cos x = _/2/2 Б) cos x/3 = _/2/2 В) cos 2 x = - _/2/2 3. А) tg x = 1 Б) tg (x - ㅠ/3) = 1 В) tg x = - 1 4. А) sin 3x = 2 Б) cos x/4 = - _/3 В) tg x (2- cos x) = 0​

Answer 1

1.

А) sin x = 1/2

x = 30° + 360°k або x = 150° + 360°k, де k - ціле число.

Б) sin 2x = 1/2

2x = 30° + 360°k або 2x = 150° + 360°k

x = 15° + 180°k або x = 75° + 180°k, де k - ціле число.

2.

А) cos x = √2/2

x = 45° + 360°k або x = 315° + 360°k, де k - ціле число.

Б) cos x/3 = √2/2

x/3 = 45° + 360°k або x/3 = 315° + 360°k

x = 135° + 1080°k або x = 945° + 1080°k, де k - ціле число.

В) cos 2x = -√2/2

2x = 135° + 360°k або 2x = 225° + 360°k

x = 67.5° + 180°k або x = 112.5° + 180°k, де k - ціле число.

3.

А) tg x = 1

x = 45° + 180°k, де k - ціле число.

Б) tg (x - π/3) = 1

x - π/3 = π/4 + πk або x - π/3 = 5π/4 + πk

x = 7π/12 + πk або x = 11π/12 + πk, де k - ціле число.

В) tg x = -1

x = 135° + 180°k, де k - ціле число.

4.

А) sin 3x = 2

Рівняння не має розв'язків, оскільки значення синуса не може перевищувати 1.

Б) cos x/4 = -√3/3

x/4 = 150° + 360°k або x/4 = 210° + 360°k

x = 600° + 1440°k або x = 840° + 1440°k, де k - ціле число.

В) tg x (2 - cos x) = 0

Розв'язками є x = 0° + 180°k або x = 60° + 180°k, де k - ціле число.

Answer 2

a) sin 2x = 1/2 To solve this equation, we can use the inverse sine function (also known as arcsine). Taking the inverse sine of both sides, we have: 2x = arcsin(1/2) x = arcsin(1/2) / 2

b) sin(x/2) = 1/2 Using the same approach, we take the inverse sine of both sides: x/2 = arcsin(1/2) x = 2 * arcsin(1/2)

a) cos x = √(2)/2 To solve this equation, we can use the inverse cosine function (also known as arccosine): x = arccos(√(2)/2)

b) cos(x/3) = √(2)/2 Taking the inverse cosine of both sides: x/3 = arccos(√(2)/2) x = 3 * arccos(√(2)/2)

c) cos 2x = -√(2)/2 Taking the inverse cosine of both sides: 2x = arccos(-√(2)/2) x = arccos(-√(2)/2) / 2

a) tan x = 1 To solve this equation, we can use the inverse tangent function (also known as arctangent): x = arctan(1)

b) tan(x - π/3) = 1 Taking the inverse tangent of both sides: x - π/3 = arctan(1) x = arctan(1) + π/3

c) tan x = -1 Using the inverse tangent function: x = arctan(-1)

a) sin 3x = 2 This equation has no real solutions since the sine function is bounded by -1 and 1.

b) cos(x/4) = -√(3)/2 Taking the inverse cosine: x/4 = arccos(-√(3)/2) x = 4 * arccos(-√(3)/2)

c) tan x (2 - cos x) = 0 This equation can be rewritten as two separate equations: tan x = 0 or (2 - cos x) = 0

For the first equation, tan x = 0, we have: x = arctan(0)

For the second equation, (2 - cos x) = 0, we have: cos x = 2 This equation has no real solutions since the cosine function is bounded by -1 and 1.