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.

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 value of 2t. The cosine function gives 0.5 at two points in the unit circle, π/3 and -π/3. But since we are looking for solutions in the interval [0, π/2], we only consider the positive solution:

2t = π/3.

Step 4: Finally, 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.

Claro, aquí tienes la solución paso a paso:

1. Comienza con la ecuación dada: $4\cos(2t) + 1 = 3$

2. Resta 1 de ambos lados de la ecuación: $4\cos(2t) = 2$

3. Divide ambos lados de la ecuación por 4: $\cos(2t) = \frac{2}{4} = \frac{1}{2}$

4. Ahora, encuentra los valores de $2t$ que satisfacen $\cos(2t) = \frac{1}{2}$. Sabemos que: $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$

   Entonces: $2t = \frac{\pi}{3}$

5. Divide ambos lados de la ecuación por 2 para encontrar $t$: $t = \frac{\pi}{6}$

6. Verifica que la solución esté en el intervalo dado $[0, \frac{\pi}{2}]$. En este caso, $\frac{\pi}{6}$ está dentro del intervalo.

Por lo tanto, la solución exacta en el intervalo $[0, \frac{\pi}{2}]$ es: $t = \frac{\pi}{6}$