91Ó°ÊÓ

Understanding the addition of angles is crucial for solving various trigonometry problems. It allows us to express the tangent of the sum of two angles in terms of the tangents of the individual angles. The formula is given by:
\[ \tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha)\tan(\beta)} \]
In the context of the given problem, the addition of angles formula is used to combine the arctangents of two separate values.

Instead of viewing the problem as dealing with two distinct arctangents being added, we can use this formula to unify them into a single expression. This is because the arctangent function represents an angle, and thus the addition of two arctangents is equivalent to the addition of two angles. When applying this concept, one must be careful to consider the domain restrictions of the tangent function, ensuring that the angles being added fall within the valid range.

Definition and Properties

The arctangent function, denoted as \( \tan^{-1} \), is the inverse of the tangent function. It returns the angle whose tangent is the given number. Therefore, if \( \tan(\theta) = x \), then \( \tan^{-1}(x) = \theta \).

A key property of the arctangent function is that it always returns values in the range \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \), or \( -90^\circ \) to \( 90^\circ \), making it a function that deals with acute and obtuse angles. This property comes into play when working with the addition of angles, as seen in the original exercise, where the result needs adjustment to be within the correct interval.

When we have the expression \( 2 \tan ^{-1}\left(\frac{3}{2}\right) + \tan^{-1}\left(\frac{12}{5}\right) = \pi \), we're effectively looking for the angle that, when transformed by the tangent function, gives us the number on the right-hand side of the equation. The process of combining the arctangents of numbers effectively angles, where the result will be an angle within the principal range.

Understanding Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables where both sides of the equations are defined. These identities are useful in simplifying expressions and solving equations.

A familiar trigonometric identity that applies in our exercise is the periodicity of the tangent function, expressed as \( \tan(\theta) = \tan(\theta + k\pi) \) for any integer \( k \). This reflects the idea that the tangent function repeats every \( \pi \) radians.

The periodic property was used to adjust the angle in the solution, recognizing that an angle plus \( \pi \) has the same tangent as the angle itself. For instance, if \( \tan(\theta) = x \), adding \( \pi \) to \( \theta \) doesn't change the tangent's value. This is why in the final step of the solution, the equation could be reinterpreted to find a corresponding angle in the principal range that satisfies the initial equation.