Solving the Trigonometric Equation: Tan 3x 3 and Its General Solutions
Trigonometric equations are a fundamental component of mathematics, particularly in the field of trigonometry. In this article, we will explore the process of solving the equation tan 3x 3. This involves understanding the concept of reference angles and the general solutions for trigonometric equations.
Understanding Trigonometric Equations
A trigonometric equation involves trigonometric functions of an angle or variable, such as sine, cosine, and tangent. The goal is to find the values of the variable (often denoted as x) that satisfy the given equation. In this case, our equation is tan 3x 3.
Solving the Equation: Tan 3x 3
The given equation is:
tan 3x 3
To solve this equation, we first need to find the value of the base angle, which is the reference angle for which the tangent is 3.
3x arctan 3
Here, the arctan 3 represents the reference angle whose tangent is 3. The arctan function returns the angle whose tangent is the given value.
General Solutions
Since trigonometric functions are periodic, the general solutions for the equation tan 3x 3 will involve not just the reference angle, but also all possible angles in the periodic interval. For the tangent function, the period is pi. Therefore, to find the general solutions, we add integer multiples of the period to the reference angle.
The general solutions can be written as:
3x arctan 3 ppi n
Where n in mathbb{Z} (the set of all integers).
Now, to isolate x, we divide both sides of the equation by 3:
x frac{arctan 3 ppi n}{3}
Thus, the general solutions for the equation tan 3x 3 are:
x frac{arctan 3 ppi n}{3}
Where n in mathbb{Z}.
Conclusion
In conclusion, solving the trigonometric equation tan 3x 3 involves understanding the concept of reference angles and recognizing the periodic nature of trigonometric functions. The general solution for this equation is x frac{arctan 3 ppi n}{3}, where n is an integer. This solution provides a complete set of values for x that satisfy the given equation.