Answer:
1. x = 45°
2. x = 330°
3. x = 105°
4. x = 30°
5. x = 60°
6. x = 30° or x = -45° (315°) or x = 45°
Step-by-step explanation:
1. For
cos x = sin x
∴ sin x/(cos x) = 1 = tan x
Hence, x = tan⁻¹1 = 45°
2. For
csc(x) + 3 = 1
∴ csc(x) = 1 - 3 = -2
Which gives sin x = -1/2
∴ x = sin⁻¹(-1/2) = -30° = 360 +(-30) = 330 °
3. For
cot(3x) = -1
∵ cot(3x) = 1/(tan(3x))
Hence, 1/(tan(3x)) = -1
∴ tan(3x) = -1
3·x = tan⁻¹(-1) = -45° = 360 + (-45) = 315°
Which gives, x = -45/3 = -15° or x = 105°
4. For
2·sin²(x) + 3·sin(x) = 2
We put sin(x) = y to get
2·y² + 3·y = 2 or 2·y² + 3·y - 2 =0
Factorizing gives
(2·y -1)(y+2) =0
∴ y = 1/2 or y = -2
That is, sin(x) = 1/2 or sin(x) = -2
Hence, x = sin⁻¹(1/2) = 30° or x = sin⁻¹(-2) = (-π/2 + 1.3·i)
∴ x = 30°
5. For
4cos²(x) = 3 we have;
cos²(x) = 3/4
cos(x) = √(3/4) = (√3)/2
∴ x = cos⁻¹((√3)/2) = 60°
6. For
4·sin³(x) + 1 = 2·sin²(x) + 2·sin(x)
We put sin(x) = y to get;
4·y³ + 1 = 2·y² + 2·y which gives;
4·y³ + 1 - (2·y² + 2·y) = 0 or 4·y³ -2·y² - 2·y + 1 = 0
Factorizing gives;
Therefore, x = 1/2 or x = -(√2)/2 or (√2)/2
Therefore, sin(x) = 1/2 or -(√2)/2 or (√2)/2
That is x = sin⁻¹(1/2) = 30 or sin⁻¹(-(√2)/2) = -45 or sin⁻¹((√2)/2) = 45.
