Question

1+2 sin^2 (πx/18)+√3cos(πx/18)

Answer 1

(πx/18)=±arccos(-√3/2)+2πk, k∈Z.
(πx/18)=±(π -arccos(√3/2))+2πk, k∈Z.
(πx/18)=±(π - (π/6))+2πk, k∈Z.
(πx/18)=± (5π/6)+2πk, k∈Z.
(x/18)= ± (5/6)+2k, k∈Z.
x=± 15+36k, k∈Z.
x=15 - наименьший положительный корень.
Если конечно нужен Наименьший положительный корень был.

Answer 2

To simplify the expression, let's work step by step:

1. Apply the trigonometric identity: sin^2(x) + cos^2(x) = 1

2. Use the identity to rewrite the expression:

   1 + 2 sin^2(πx/18) + √3 cos(πx/18)

   = 1 + 2(sin^2(πx/18) + cos^2(πx/18)) + √3 cos(πx/18)

3. Apply the trigonometric identity: sin^2(x) + cos^2(x) = 1

   = 1 + 2(1) + √3 cos(πx/18)

4. Simplify:

   = 1 + 2 + √3 cos(πx/18)

5. Combine the constants:

   = 3 + √3 cos(πx/18)

So, the simplified expression is: 3 + √3 cos(πx/18)