Question 1210192
Let's solve the trigonometric equation step by step.

**1. Apply Trigonometric Identities**

* We have the equation: sin(4A) = sin(A) + sin(2A)
* Apply double angle formula: sin(2A) = 2sin(A)cos(A)
* Apply double angle formula: sin(4A) = 2sin(2A)cos(2A)
* Substitute sin(2A) into sin(4A): sin(4A) = 2(2sin(A)cos(A))cos(2A) = 4sin(A)cos(A)cos(2A)
* So the equation becomes: 4sin(A)cos(A)cos(2A) = sin(A) + 2sin(A)cos(A)

**2. Simplify the Equation**

* 4sin(A)cos(A)cos(2A) = sin(A)(1 + 2cos(A))

**3. Consider sin(A) = 0**

* If sin(A) = 0, then A = 0° or 180°.
* Since we are looking for an acute angle, A = 0° is a solution.
* However, we are looking for an acute angle, so we need to consider sin(A) != 0

**4. Divide by sin(A)**

* Assume sin(A) ≠ 0. Divide both sides by sin(A):
* 4cos(A)cos(2A) = 1 + 2cos(A)

**5. Apply Double Angle Formula for cos(2A)**

* cos(2A) = 2cos²(A) - 1
* Substitute into the equation: 4cos(A)(2cos²(A) - 1) = 1 + 2cos(A)
* 8cos³(A) - 4cos(A) = 1 + 2cos(A)
* 8cos³(A) - 6cos(A) - 1 = 0

**6. Solve for cos(A)**

* Let x = cos(A). Then we have the cubic equation:
* 8x³ - 6x - 1 = 0

* We can use numerical methods or trigonometric identities to solve this.
* Using a cubic equation solver, we find one real root approximately: x ≈ 0.8365

* Let A = arccos(0.8365).
* A ≈ 33.24°

**7. Verify the Solution**

* Check if A ≈ 33.24° satisfies the original equation.

* sin(4A) = sin(4 * 33.24) ≈ sin(132.96) ≈ 0.7313
* sin(A) + sin(2A) = sin(33.24) + sin(66.48) ≈ 0.5481 + 0.9168 ≈ 1.4649
* This is incorrect.

**8. Finding the Exact Solution**

* There is another method to solve it. Let $x = cosA$. Then $cos2A = 2x^2-1$.
* The equation $8x^3-6x-1=0$ can be transformed into the form $4x^3-3x = 1/2$.
* $cos(3A) = 4cos^3A-3cosA$.
* Therefore, $cos(3A) = 1/2$.
* $3A = 60$.
* $A = 20$.
* Let's test if A=20 is a solution.
* sin(4A) = sin(80).
* sin(A) + sin(2A) = sin(20)+sin(40).
* sin(80) = sin(20)+sin(40).
* sin(80) = sin(60-40)+sin(60-20).
* sin(80) = sin(60)cos(40)-cos(60)sin(40)+sin(60)cos(20)-cos(60)sin(20).
* sin(80) = sqrt(3)/2*cos(40)-1/2*sin(40)+sqrt(3)/2*cos(20)-1/2*sin(20).

**9. Conclusion**

The acute angle A that satisfies the equation is 20 degrees.