Question

Arcsin((5x-1)/3)+2arccos((5x-1)/3)=5пи/6​

Answer 1

Ответ:х=0.5

Пошаговое объяснение:arcsin((5x-1)/3)+2arccos((5x-1)/3)=5π/6​

arccos((5x-1)/3)=π/2-arcsin((5x-1)/3), при условии  -1≤(5x-1)/3≤1;

-3≤(5x-1)≤3; -2≤5x≤4;-0.4≤x≤0.8;

arcsin((5x-1)/3)+2*(π/2-arcsin((5x-1)/3))=5π/6​

π-arcsin((5x-1)/3)=5π/6​

arcsin((5x-1)/3)=π/6​

(5x-1)/3=0.5

5x-1=1.5

5x=2.5

x=2.5/5

х=0.5∈ОДЗ.

Answer 2

To solve the equation:

arcsin((5x - 1)/3) + 2arccos((5x - 1)/3) = 5π/6,

we can use the fact that the sum of the arcsin and arccos of the same value equals π/2. Let's solve it step by step:

1. Let's rewrite the equation using this property: arcsin((5x - 1)/3) + arccos((5x - 1)/3) + arccos((5x - 1)/3) = 5π/6.

2. Now, we can replace one of the arccos terms with π/2 - arcsin, using the fact mentioned earlier: arcsin((5x - 1)/3) + arccos((5x - 1)/3) + (π/2 - arcsin((5x - 1)/3)) = 5π/6.

3. Simplify the equation: arcsin((5x - 1)/3) + arcsin((5x - 1)/3) + (π/2 - arcsin((5x - 1)/3)) = 5π/6.

4. Combine like terms: 2arcsin((5x - 1)/3) + (π/2 - arcsin((5x - 1)/3)) = 5π/6.

5. Distribute the negative sign: 2arcsin((5x - 1)/3) + π/2 - arcsin((5x - 1)/3) = 5π/6.

6. Combine like terms: arcsin((5x - 1)/3) + π/2 = 5π/6.

7. Subtract π/2 from both sides: arcsin((5x - 1)/3) = 5π/6 - π/2.

8. Simplify the right side: arcsin((5x - 1)/3) = (5π - 3π)/6.

9. Further simplify: arcsin((5x - 1)/3) = 2π/6.

10. Simplify the right side: arcsin((5x - 1)/3) = π/3.

11. Apply the arcsin function to both sides: (5x - 1)/3 = sin(π/3).

12. Simplify the right side: (5x - 1)/3 = √3/2.

13. Multiply both sides by 3: 5x - 1 = 3√3/2.

14. Add 1 to both sides: 5x = 3√3/2 + 1.

15. Divide both sides by 5: x = (3√3/2 + 1)/5.

Therefore, the solution to the equation arcsin((5x - 1)/3) + 2arccos((5x - 1)/3) = 5π/6 is:

x = (3√3/2 + 1)/5.