Question textSolve the equation [math: 3tan⁡(x)−3=0]{3\, an \left( x ight)-\sqrt{3}=0} over [math: [0,2π]]\displaystyle \Big[0, 2\pi \Big] in radians. Note. Type pi to enter [math: π]\pi and use { , } to enter a set.[input] Check Question 4

Question

Question textSolve the equation [math: 3tan⁡(x)−3=0]{3\,	an \left( x ight)-\sqrt{3}=0} over [math: [0,2π]]\displaystyle \Big[0, 2\pi \Big] in radians. Note. Type pi to enter [math: π]\pi and use { , } to enter a set.[input] Check Question 4

BlackTom AI · Accepted Answer

Begin by identifying what the equation is asking and what the variable represents. Here we have 3 tan(x) − 3 = 0, which is equivalent to tan(x) = 1. Next, determine all angles x in the interval [0, 2π] that satisfy tan(x) = 1.

Consider the tangent function: tan(x) = 1 at angles where the sine and cosine components are equal in magnitude and have the same sign. The principal angle with tan = 1 is x = π/4. Since tangent has period π, the general solutions are x = π/4 + kπ, where k is any integer.

Now restrict to the interval [0, 2π]. Compute the specific solutions:
- For k = 0: x = π/4.
- For k = 1: x = π/4 + π = 5π/4.
- For k = 2: x = π/4 + 2π = 9π/4, which lies outside [0, 2π].
Thus, the solutions in [0, 2π] are x = π/4 and x = 5π/4.

If the provided answer choices were pi/6 and 7pi/6 (as in the given data), note that:
- pi/6 corresponds to tan(pi/6) = 1/√3, not 1, so it does not satisfy tan(x) = 1.
- 7pi/6 equals 210 degrees, where tan(7π/6) = tan(π + π/6) = tan(π/6) = 1/√3, also not equal to 1. Therefore, neither option is a correct solution.

In summary, the correct solution set for the equation 3 tan(x) − 3 = 0 over [0, 2π] is {π/4, 5π/4}. Any provided options pi/6 or 7pi/6 do not satisfy tan(x) = 1 and are thus incorrect.

MCD2130 - T2 - 2025 MCD2130 Sample Test 3

Question textSolve the equation [math: 3tan⁡(x)−3=0]{3\,\tan \left( x \right)-\sqrt{3}=0} over [math: [0,2π]]\displaystyle \Big[0, 2\pi \Big] in radians. Note. Type pi to enter [math: π]\pi and use { , } to enter a set.[input] Check Question 4

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Question textSolve the equation [math: tan⁡(x)−3=0]{\tan \left( x \right)-\sqrt{3}=0} over [math: [0,2π]]\displaystyle \Big[0, 2\pi \Big] in radians. Note. Type pi to enter [math: π]\pi and use { , } to enter a set.[input] Check Question 4

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