91Ó°ÊÓ

The tangent function, often abbreviated as tan, is one of the primary trigonometric functions. It relates an angle of a right triangle to the ratio of the opposite side over the adjacent side. This function is periodic and repeats its values over specific intervals.

It is defined as:

• \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)

Due to its definition, the tangent function can have undefined values wherever the cosine function is zero, leading to vertical asymptotes in its graph. This typically occurs at angles like \( \frac{\pi}{2}, \frac{3\pi}{2} \), etc.
Beyond its definition, the tangent function has a period of \( \pi \). This means that every \( \pi \) units, the function returns to its starting value. Hence, when solving equations involving \( \tan(x) \), we only consider solutions within one full cycle or within a desired interval, as done in the original exercise. This periodicity is crucial in finding all solutions in a given range, as it helps encapsulate all possible angles that satisfy the given equation.

In mathematics, many trigonometric equations are given in quadratic form, especially when involving functions like sine, cosine, and tangent. Recognizing a quadratic form in trigonometric equations can simplify solving processes by borrowing techniques from algebra.

A basic quadratic equation takes the form:

• \( ax^2 + bx + c = 0 \)

Similarly, a trigonometric equation in quadratic form might look like \( a(\tan x)^2 + b(\tan x) + c = 0 \), using tangent as an example.
To solve these, we can apply standard algebraic techniques: rearranging the equation, factoring, or using the quadratic formula. In the exercise, the original equation \( 3 \tan^2 x - 9 = 0 \) is already almost in a quadratic format, which allows us to conveniently solve by isolating \( \tan^2 x \). Recognizing this quadratic structure is essential, as it allows us to address more complex trigonometric problems efficiently.

The Unit Circle is a fundamental tool in trigonometry that helps visualize and solve equations involving trigonometric functions, like sine, cosine, and tangent. It is a circle with a radius of 1, centered at the origin of the coordinate plane.

On the unit circle, every point \((x, y)\) represents:

• \( x = \cos(\theta) \)
• \( y = \sin(\theta) \)

For tangent, the unit circle provides a geometrical interpretation: the tangent of an angle is the y-coordinate divided by the x-coordinate (or \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)).
Solutions for tangent involve identifying angles where this division yields the values computed from the trigonometric equation. For example, if \( \tan(x) = \sqrt{3} \), we search for angles on the unit circle where \( \frac{y}{x} = \sqrt{3} \).
The unit circle becomes particularly useful when determining which quadrants a solution lies in, especially since the sign of the tangent changes between quadrants. This understanding is essential when finding multiple solutions for the tangent over an interval, as seen in the original exercise where solutions spanned different quadrants.