Answer :

To solve the equation cos⁡(3)−2=−52cos(3)−2=−25​ on the interval [0,π)[0,π), we first need to simplify the left side:cos⁡(3)−2=−52cos(3)−2=−25

​Next, we need to isolate the cosine term:cos⁡(3)=−52+2cos(3)=−25​+2cos⁡(3)=−52+42cos(3)=−25​+24​cos⁡(3)=−12cos(3)=−21​

Now, we need to find the angle θθ in the interval [0,π)[0,π) such that cos⁡(θ)=−12cos(θ)=−21​.

In the interval [0,π)[0,π), the cosine function is negative in the second and third quadrants. The reference angle for cos⁡−1(−12)cos−1(−21​) is π33π​, which corresponds to the angle in the second quadrant where the cosine is negative. Therefore, the solution is θ=π−π3=2π3θ=π−3π​=32π​.

So, the solution to the equation cos⁡(3)−2=−52cos(3)−2=−25​ on the interval [0,π)[0,π) is 2π332π​.

lukasgarciafresco

Answer:

To solve the equation [tex]\( \cos(3) - 2 = -\frac{5}{2} \)[/tex] on the interval [tex]\([0,\pi)\)[/tex], we'll first find the value of \( \cos(3) \) and then check if it satisfies the equation.

Using a calculator or trigonometric table, we find that [tex]\( \cos(3) \approx -0.989 \).[/tex]

Now, let's substitute this value into the equation:

[tex]\[ \cos(3) - 2 = -\frac{5}{2} \][/tex]

[tex]\[ -0.989 - 2 = -\frac{5}{2} \][/tex]

[tex]\[ -2.989 = -\frac{5}{2} \][/tex]

This equation is not true, so [tex]\( \cos(3) - 2 = -\frac{5}{2} \)[/tex] is not satisfied on the interval [tex]\([0,\pi)\)[/tex]. Therefore, there is no solution to the equation on this interval.

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