tan(2x)=root3 over 3
To solve the equation tan(2x) = √3/3, we can use the trigonometric identity for double angles:
tan(2x) = 2tan(x) / (1 - tan^2(x))
Let's solve this equation step by step:
Step 1: Rewrite the given equation using the double-angle identity:
2tan(x) / (1 - tan^2(x)) = √3/3
Step 2: Multiply both sides of the equation by (1 - tan^2(x)) to eliminate the denominator:
2tan(x) = (√3/3)(1 - tan^2(x))
Step 3: Distribute on the right side of the equation:
2tan(x) = (√3/3) - (√3/3)tan^2(x)
Step 4: Move all terms to one side of the equation:
2tan(x) - (√3/3) + (√3/3)tan^2(x) = 0
Step 5: Combine like terms:
2tan(x) + (√3/3)(√3/3)tan^2(x) - (√3/3) = 0
Step 6: Simplify (√3/3)(√3/3) to (√3/9):
2tan(x) + (√3/9)tan^2(x) - (√3/3) = 0
Step 7: Multiply the entire equation by 9 to eliminate the fraction:
18tan(x) + √3tan^2(x) - 3√3 = 0
Step 8: Rearrange the equation:
√3tan^2(x) + 18tan(x) - 3√3 = 0
Now, we have a quadratic equation in terms of tan(x). To solve this equation, we can use the quadratic formula:
tan(x) = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = √3, b = 18, and c = -3√3.
Step 9: Substitute the values of a, b, and c into the quadratic formula and simplify:
tan(x) = (-18 ± √(18^2 - 4(√3)(-3√3))) / (2√3)
tan(x) = (-18 ± √(324 + 36)) / (2√3)
tan(x) = (-18 ± √(360)) / (2√3)
Step 10: Simplify the expression further:
tan(x) = (-18 ± √(36 * 10)) / (2√3)
tan(x) = (-18 ± 6√10) / (2√3)
tan(x) = (-9 ± 3√10) / √3
Step 11: Rationalize the denominator by multiplying the numerator and denominator by √3:
tan(x) = [(-9 ± 3√10) * √3] / (√3 * √3)
tan(x) = (-9√3 ± 3√30) / 3
tan(x) = -3√3 ± √30
Therefore, the solutions for x are:
x = arctan(-3√3 + √30) and x = arctan(-3√3 - √30).
To find the value of x in the equation tan(2x) = √3/3, we can use the inverse tangent function (also known as arctan or tan^(-1)). The inverse tangent function allows us to find the angle whose tangent is a given value.
Here's how we can solve for x:
Step 1: Take the inverse tangent of both sides of the equation.
arctan(tan(2x)) = arctan(√3/3)
The inverse tangent of a tangent function cancels out, so we have:
2x = arctan(√3/3)
Step 2: Simplify the right side of the equation.
The value of arctan(√3/3) can be found using a calculator or by recognizing the special angle relationships. In this case, we can identify that arctan(√3/3) is equal to π/6 or 30 degrees.
Therefore, 2x = π/6 or 2x = 30 degrees.
Step 3: Solve for x.
To find the value of x, we divide both sides of the equation by 2:
x = π/12 or x = 15 degrees.
So the solution to the equation tan(2x) = √3/3 is x = π/12 or x = 15 degrees.