Question

What are all the exact solutions of -3tan²(x)+1=0? Give your answer in radians.

Answer 1

The exact solutions of the equation -3tan²(x) + 1 = 0 in radians are all values of the form x = π/6 + kπ and x = 5π/6 + kπ, where k is any integer.

Step-by-step explanation:

To find all the exact solutions of the equation -3tan²(x) + 1 = 0, we first solve for tan²(x):

1. Isolate the tangent term: tan²(x) = 1/3.
2. Take the square root of both sides: tan(x) = ±1/√3.
3. Now we find x by using the arctangent function: x = arctan(±1/√3).
4. The positive solution gives x = arctan(1/√3) which is π/6. Since tan(x) is periodic with a period of π, all solutions for the positive square root will be x = π/6 + kπ, where k is an integer.
5. The negative solution gives x = arctan(-1/√3) which is 5π/6. All solutions for the negative square root will be x = 5π/6 + kπ, where k is an integer.

Therefore, the exact solutions of the equation in radians are all values of the form x = π/6 + kπ and x = 5π/6 + kπ, where k is any integer.

It's important to check these solutions to ensure they are reasonable and fit within the constraints of the original question. In this case, our solutions are consistent with the properties of the tangent function.

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