Solve the trigonometric equation4cos(2𝑡)+1=3to find the exact solution on the interval [0,𝜋2]. Give your answer in radians.

Sure, let's solve the trigonometric equation step by step.

The given equation is 4cos(2t) + 1 = 3.

Step 1: Subtract 1 from both sides of the equation to isolate the cosine function:

4cos(2t) = 3 - 1 4cos(2t) = 2

Step 2: Divide both sides of the equation by 4 to solve for cos(2t):

cos(2t) = 2/4 cos(2t) = 0.5

Step 3: Now, we need to find the angle whose cosine is 0.5. We know that cos(π/3) = 0.5 and cos(-π/3) = 0.5. But since we are looking for solutions in the interval [0, π/2], we only consider π/3.

So, 2t = π/3

Step 4: Divide both sides of the equation by 2 to solve for t:

t = π/6

So, the solution to the equation 4cos(2t) + 1 = 3 in the interval [0, π/2] is t = π/6.