91Ó°ÊÓ

Inverse trigonometric functions help us find angles when given trigonometric ratios. They are often denoted with a superscript "+". In our exercise, we encounter the inverse tangent function, written as \( \tan^{-1} \). This function lets us determine the angle whose tangent is a specific value.
For example, if \( y = \tan(x) \), then \( x = \tan^{-1}(y) \). This concept is critical in solving equations where the unknown is inside a trigonometric function. By converting the process, we retrieve the angle and further work towards a solution.
In the context of the exercise, \( \tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4} \), suggesting that we are looking for a specific condition where the sum of angles equals \( \frac{\pi}{4} \), a classic angle whose tangent value we know well.

The addition formula for tangent is a tool used to simplify expressions involving the sum of two inverse tangent values. The formula is given by \( \tan^{-1}(u) + \tan^{-1}(v) = \tan^{-1}\left(\frac{u+v}{1-uv}\right) \) when the product \( uv < 1 \).
This formula can simplify the expression \( \tan^{-1}(2x) + \tan^{-1}(3x) \), allowing us to express it as a single inverse tangent term. This step is crucial because it condenses the equation, making it easier to solve.
Using the formula, we translate multiple inverse tangent expressions into a manageable equation like \( \tan^{-1}\left(\frac{5x}{1-6x^2}\right) \), which simplifies the process and helps us isolate the variable \( x \).

Solving equations involving trigonometric functions often requires a clear strategy and application of identities. Here’s how we tackled the problem.

• After simplifying using the addition formula, we are left with \( \tan\left(\frac{\pi}{4}\right) = 1 \), a familiar trigonometric value.
• This equates our fraction \( \frac{5x}{1-6x^2} \) to 1.
• Next, solve the equation by setting the numerator equal to the denominator: \( 5x = 1 - 6x^2 \).
• Rearrange and solve the quadratic equation. You deduce it down to the solution \( x = \sqrt{\frac{1}{6}} \).

Throughout the process, simplifying the equation step-by-step, using known trigonometric values, and standard algebraic techniques plays a vital role in isolating \( x \) and finding the answer.