Problem 83 - A Graphical Approach to Precalculus with Limits

Question

Give the exact real number value of each expression. Do not use a calculator. $$\sin \left(2 \tan ^{-1} \frac{12}{5}\right)$$

Step-by-Step Solution

Answer

The exact value of the expression is \( \frac{120}{169} \).

Step 1: Understand the Inverse Tangent Function

The expression involves \( \tan^{-1} \frac{12}{5} \) which is the angle \( \theta \) whose tangent is \( \frac{12}{5} \). This means \( \tan \theta = \frac{12}{5} \).

Step 2: Use Trigonometric Identity for Sine of Double Angle

The problem requires \( \sin(2\theta) \) where \( \theta = \tan^{-1} \frac{12}{5} \). We use the identity: \[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \].

Step 3: Find \( \sin \theta \) and \( \cos \theta \)

Knowing that \( \tan \theta = \frac{12}{5} \), we can create a right triangle with opposite side 12, adjacent side 5, which gives the hypotenuse as \( \sqrt{12^2 + 5^2} = \sqrt{169} = 13 \). Thus, \( \sin \theta = \frac{12}{13} \) and \( \cos \theta = \frac{5}{13} \).

Step 4: Calculate \( \sin(2\theta) \) Using the Double Angle Identity

Substitute \( \sin \theta \) and \( \cos \theta \) into the double angle formula: \[ \sin(2\theta) = 2 \times \frac{12}{13} \times \frac{5}{13} = 2 \times \frac{60}{169} = \frac{120}{169} \].

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are incredibly useful when working with angles and trigonometric ratios. They allow us to find an angle when we know the trigonometric ratio associated with that angle. For instance, when you see the expression \( \tan^{-1} \frac{12}{5} \), it signifies the angle \( \theta \) such that \( \tan \theta = \frac{12}{5} \).
This is essential because it helps transform a ratio back into an angle, which offers a starting point for many trigonometric equations.

• The inverse of \( \tan \) is often referred to as \( \tan^{-1} \) or arctangent.
• It determines an angle in a right triangle based on the tangent value.
• Understanding these functions helps bridge the gap between straight-angle measurements and their various trigonometric forms.

In this exercise, we've used \( \tan^{-1} \frac{12}{5} \) to understand the angle whose tangent is \( \frac{12}{5} \), setting us up for further calculations.

Double Angle Formulas

The double angle formulas are a powerful tool in trigonometry that help in simplifying expressions involving trigonometric functions of double angles. In our current problem, we are tasked with finding \( \sin(2\theta) \) using the double angle formula for sine:
\[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \]These formulas come in handy when an angle can be expressed as twice another angle, and you need to compute trigonometric functions on this new angle.

• They help break down complex trigonometric expressions into more manageable parts.
• They are derived from the angle addition formulas and are part of the foundation of trigonometric identities.
• By using these formulas, calculations that seem complex at first can become quick and efficient.

In the context of solving \( \sin \left( 2 \tan^{-1} \frac{12}{5} \right) \), the double angle formula simplifies the computation by using \( \sin \theta \) and \( \cos \theta \) found from our right triangle.

Right Triangle Trigonometry

Understanding right triangle trigonometry is foundational for solving many trigonometric problems, especially those involving inverse functions and identities. Here, we deal with an angle \( \theta \) found by \( \tan^{-1} \frac{12}{5} \), which translates to a right triangle where the opposite side is 12 and the adjacent side is 5.
We use the Pythagorean theorem to find the hypotenuse, forming a complete picture of the triangle.

• The hypotenuse \( c \) is calculated as \( \sqrt{12^2 + 5^2} = 13 \).
• The sine and cosine ratios are retrieved from this triangle: \( \sin \theta = \frac{12}{13} \) and \( \cos \theta = \frac{5}{13} \).
• This understanding is critical in using the double angle formula since both \( \sin \theta \) and \( \cos \theta \) are required for the calculation.

By mastering right triangle trigonometry, you create strong foundational skills that enable the use of more complex mathematical identities.