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The tangent function is one of the primary trigonometric functions, commonly used in mathematics to compare the ratios of the sides of right triangles. It is defined using the sine and cosine functions. For any angle \( \theta \), \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). The inverse tangent, denoted as \( \tan^{-1}(x) \) or \( \arctan(x) \), is the function that gives the angle when the tangent is known.
Understanding the tangent function and its inverse is essential when working with angles and solving trigonometric equations involving inverse functions. In problems involving inverse tangents like \( \tan^{-1}(2x) + \tan^{-1}(3x) = \frac{3 \pi}{4} \), key formulas can simplify the process:

• Sum formula for inverse tangent: \[ \tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1}\left(\frac{a + b}{1 - ab}\right) \quad \text{if } ab < 1 \]

This formula helps in transforming complex expressions into simpler forms to easily find solutions.

Quadratic equations are fundamental to algebra, represented in the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. These equations are solved using various methods, including factoring, completing the square, and the quadratic formula:

• Quadratic Formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In our exercise, after simplifying the trigonometric equation, we derived a quadratic equation: \( 6x^2 + 5x + 1 = 0 \).
It's crucial to recognize the components of a quadratic equation, as they determine the method to find "x". The discriminant \( b^2 - 4ac \) tells us the nature of the roots. A positive discriminant indicates real and distinct roots, zero suggests real and identical roots, and a negative discriminant implies complex roots.

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the involved variables. These identities are instrumental in simplifying expressions and solving equations in trigonometry. Some essential identities include:

• Pythagorean Identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Angle Sum Identities: such as \( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \)
• Double Angle Identities: e.g., \( \tan(2\theta) = \frac{2\tan(\theta)}{1-\tan^2(\theta)} \)

These identities allow for the conversion of complex trigonometric expressions into more manageable forms. For instance, formulas like the angle sum identity for tangent are necessary to derive our exercise's final equation. Recognizing and applying these identities effectively helps streamline your problem-solving process.