Answer:
3 = X
Step-by-step explanation:
To find the values of x that satisfy the equation 9cot^2(x) = 3 in the interval [0, 2], we can follow these steps:
1. Rewrite the equation using the identity cot^2(x) = 1/tan^2(x). This gives us 9/(tan^2(x)) = 3.
2. Simplify the equation by multiplying both sides by tan^2(x). This yields 9 = 3tan^2(x).
3. Divide both sides by 3 to isolate tan^2(x). We now have tan^2(x) = 3/9, which simplifies to tan^2(x) = 1/3.
4. Take the square root of both sides to find tan(x). The positive and negative square roots must be considered, so we have tan(x) = ±√(1/3).
5. Use the inverse tangent function (tan^(-1)) to find the values of x. Taking the inverse tangent of both sides, we get x = tan^(-1)(±√(1/3)).
6. Use a calculator to find the principal value of tan^(-1)(±√(1/3)). The principal value of tan^(-1)(±√(1/3)) is approximately ±0.615.
7. Since we are looking for values of x in the interval [0, 2], the only solution that falls within this range is x = 0.615.
Therefore, the only value of x in the interval [0, 2] that satisfies the equation 9cot^2(x) = 3 is x = 0.615.
