91Ó°ÊÓ

Trigonometric identities are fundamental tools in solving trigonometric equations and simplifying expressions. They are equalities that involve trigonometric functions and are true for all values of the variables where both sides of the equality are defined. One key identity is the Pythagorean identity, which states that for any angle \(x\), \(\sin^2 x + \cos^2 x = 1\). This basic relationship allows us to express one trigonometric function entirely in terms of another, leading to various other identities.

Another useful set of identities includes the reciprocal identities like \(\sin x = 1/\csc x\) and \(\cos x = 1/\sec x\), and the tangent function which itself can be expressed as a quotient identity: \(\tan x = \sin x/\cos x\). Understanding these relationships is crucial for transforming complex trigonometric equations into a more manageable form, as seen in the exercise where \(\sin x/\cos x\) is simplified to \(\tan x\).

When approaching trigonometric problems, identifying the most applicable identity can simplify the calculation and solution process. For example, converting a sine function to a tangent function, as in the given exercise, makes it possible to isolate the variable more directly.

Inverse trigonometric functions reverse the action of the original trigonometric functions. For instance, if \(\sin x = y\), then \(\arcsin y = x\), provided \(x\) is in the domain of \(\arcsin y\). There are six main inverse trigonometric functions corresponding to the six primary trigonometric functions — \(\arcsin\), \(\arccos\), \(\arctan\), \(\arccsc\), \(\arcsec\), and \(\arccot\).

These functions are extremely useful when solving for angles given the value of a trigonometric function. For example, if you know the value of the tangent function and are tasked with finding the angle, you would use the \(\arctan\) function as seen in the step-by-step solution provided in the original exercise. When using inverse trigonometric functions, it is important to consider the range of the function as they are normally restricted to principal values to be uniquely defined.

The tangent function, denoted \(\tan x\), represents the ratio of \(\sin x\) to \(\cos x\) and is one of the primary trigonometric functions. It is particularly valuable when dealing with right triangles as it relates the opposite side to the adjacent side. The tangent function has a period of \(\pi\), meaning it repeats its values every \(\pi\) radians, and it tends to \(\pm\infty\) as it approaches the odd multiples of \(\pi/2\).

The arctangent function, or \(\arctan x\), is the inverse of the tangent function. It provides the angle whose tangent is a given number. Since the tangent function is not one-to-one, its domain must be restricted to \( (-\pi/2, \pi/2) \) for the function \(\arctan x\) to have a unique value. In the context of the exercise, when you have an equation like \(\tan x = \frac{3}{4} \) and need to find \(x\), you apply the \(\arctan\) function to both sides of the equation. This gives \( x = \arctan(\frac{3}{4}) \) which can be evaluated using a scientific calculator or trigonometric table to find the specific angle.