91Ó°ÊÓ

Inverse trigonometric functions play a crucial role in understanding relationships between angles and trigonometric ratios. The notation \( \tan^{-1} x \) is used to denote the angle whose tangent is \( x \). This is why \( \tan^{-1} \frac{3}{4} \) determines an angle for which the tangent is \( \frac{3}{4} \). Similarly, \( \tan^{-1} \frac{12}{5} \) gives us another angle where the tangent equals \( \frac{12}{5} \).

• The function \( \tan^{-1} \) serves as the inverse of the tangent function.
• It is important to understand the range of inverse functions; for \( \tan^{-1} \), the angle is typically restricted to \((-\frac{\pi}{2}, \frac{\pi}{2})\) radians.

When you encounter an expression like \( \tan^{-1}(x) \), always remember it refers to finding the angle linked to the given tangent value. This background knowledge is key when solving problems that incorporate these trigonometric functions.

Simplifying fractions is a vital skill in mathematics, especially when dealing with trigonometric expressions. When adding fractions such as \( \frac{3}{4} \) and \( \frac{12}{5} \), we need a common denominator to carry out the addition. The common denominator for 4 and 5 is 20, which standardizes the fractions to \( \frac{15}{20} \) and \( \frac{48}{20} \), respectively.

• To find a common denominator, identify the least common multiple of the denominators.
• Convert each fraction to an equivalent form with this common denominator before performing the addition or subtraction.

After adding these fractions, we simplify the result by combining the numerators. Another part of simplification involves multiplying fractions and reducing them to their simplest form, as seen when calculating \( \frac{3}{4} \cdot \frac{12}{5} = \frac{36}{20} = \frac{9}{5}\). This reduction makes calculations more manageable and provides a clearer solution.

The angle addition formula is a fundamental theorem in trigonometry utilized to find the tangent of the sum of two angles. According to this formula:\[\tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}\]This formula is quite powerful because it allows you to compute new angle values from known tangent values of individual angles. In this exercise, \( a = \tan^{-1} \frac{3}{4} \) and \( b = \tan^{-1} \frac{12}{5} \). Substituting these into the formula gives the expression:\[\tan \left( \tan^{-1} \frac{3}{4} + \tan^{-1} \frac{12}{5} \right) = \frac{\frac{3}{4} + \frac{12}{5}}{1 - \frac{3}{4} \cdot \frac{12}{5}}\]

• Make sure you simplify both the numerator and the denominator independently for clarity.
• This formula is invaluable when handling trigonometrical expressions with angle sums.

Once the simplification process is complete, you can substitute back into the expression to find the exact value. Understanding the angle addition concept clarifies how different trigonometric identities work together.