Evaluating \(\sin \left(2 \tan^{-1} \frac{5}{12} \right)\) using Trigonometry

The problem asks us to find the value of a trigonometric expression involving an inverse trigonometric function and a double angle.

The expression is \(\sin \left(2 \tan^{-1} \frac{5}{12} \right)\).

Let's use a substitution to simplify this expression. Let \(\theta = \tan^{-1} \frac{5}{12}\).

This means that \(\tan \theta = \frac{5}{12}\).

The original expression becomes \(\sin(2\theta)\).

We need to find the value of \(\sin(2\theta)\) given that \(\tan \theta = \frac{5}{12}\).

We can use the double angle formula for sine, which relates \(\sin(2\theta)\) to \(\tan \theta\):

\(\sin(2\theta) = \frac{2 \tan \theta}{1 + \tan^2 \theta}\)

Now, substitute the value of \(\tan \theta = \frac{5}{12}\) into this formula:

\(\sin(2\theta) = \frac{2 \left(\frac{5}{12}\right)}{1 + \left(\frac{5}{12}\right)^2}\)

Let's calculate the numerator and the denominator separately.

Numerator: \(2 \times \frac{5}{12} = \frac{10}{12} = \frac{5}{6}\)

Denominator: \(1 + \left(\frac{5}{12}\right)^2 = 1 + \frac{5^2}{12^2} = 1 + \frac{25}{144}\)

To add 1 and \(\frac{25}{144}\), we find a common denominator:

\(1 + \frac{25}{144} = \frac{144}{144} + \frac{25}{144} = \frac{144 + 25}{144} = \frac{169}{144}\)

Now substitute the simplified numerator and denominator back into the expression for \(\sin(2\theta)\):

\(\sin(2\theta) = \frac{\frac{5}{6}}{\frac{169}{144}}\)

To divide by a fraction, we multiply by its reciprocal:

\(\sin(2\theta) = \frac{5}{6} \times \frac{144}{169}\)

We can simplify this expression by canceling out common factors. Both 6 and 144 are divisible by 6. \(144 \div 6 = 24\).

\(\sin(2\theta) = \frac{5}{1} \times \frac{24}{169}\)

\(\sin(2\theta) = \frac{5 \times 24}{1 \times 169}\)

\(\sin(2\theta) = \frac{120}{169}\)

Thus, the value of \(\sin \left(2 \tan^{-1} \frac{5}{12} \right)\) is \(\frac{120}{169}\).

Step-by-Step Calculation

1. Identify the expression: \(\sin \left(2 \tan^{-1} \frac{5}{12} \right)\).
2. Let \(\theta = \tan^{-1} \frac{5}{12}\), which implies \(\tan \theta = \frac{5}{12}\).
3. Rewrite the expression in terms of \(\theta\): \(\sin(2\theta)\).
4. Use the double angle formula \(\sin(2\theta) = \frac{2 \tan \theta}{1 + \tan^2 \theta}\).
5. Substitute \(\tan \theta = \frac{5}{12}\): \( \sin(2\theta) = \frac{2 \left(\frac{5}{12}\right)}{1 + \left(\frac{5}{12}\right)^2} \).
6. Calculate the numerator: \(2 \times \frac{5}{12} = \frac{10}{12} = \frac{5}{6}\).
7. Calculate the denominator: \(1 + \left(\frac{5}{12}\right)^2 = 1 + \frac{25}{144} = \frac{144 + 25}{144} = \frac{169}{144}\).
8. Divide the numerator by the denominator: \( \frac{\frac{5}{6}}{\frac{169}{144}} = \frac{5}{6} \times \frac{144}{169} \).
9. Simplify the multiplication: \( \frac{5}{6} \times \frac{144}{169} = \frac{5 \times 24}{169} = \frac{120}{169} \).
10. The result is \(\frac{120}{169}\).

Comparing with Options

The calculated value is \(\frac{120}{169}\). Let's compare this with the given options:

• Option 1: \( \frac{120}{169} \)
• Option 2: \( -\frac{120}{169} \)
• Option 3: \( \frac{169}{120} \)
• Option 4: \( -\frac{169}{120} \)

Our calculated value matches Option 1.

Revision Table: Key Trigonometric Concepts

Additional Information: Alternative Approach using a Right Triangle

We can also solve this problem by constructing a right triangle. If \(\theta = \tan^{-1} \frac{5}{12}\), then \(\tan \theta = \frac{5}{12}\).

In a right triangle, \(\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}\). So, we can have a triangle with opposite side = 5 and adjacent side = 12.

Using the Pythagorean theorem, the hypotenuse \(h\) is \(h = \sqrt{\text{Opposite}^2 + \text{Adjacent}^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13\).

Now we know the sides of the right triangle are 5, 12, and 13.

From this triangle, we can find \(\sin \theta\) and \(\cos \theta\):

\(\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{5}{13}\)

\(\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{12}{13}\)

We need to find \(\sin(2\theta)\). The double angle formula for sine is \(\sin(2\theta) = 2 \sin \theta \cos \theta\).

Substitute the values of \(\sin \theta\) and \(\cos \theta\):

\(\sin(2\theta) = 2 \times \frac{5}{13} \times \frac{12}{13}\)

\(\sin(2\theta) = 2 \times \frac{5 \times 12}{13 \times 13}\)

\(\sin(2\theta) = 2 \times \frac{60}{169}\)

\(\sin(2\theta) = \frac{120}{169}\)

This alternative method using a right triangle confirms the result obtained using the double angle formula in terms of tangent.

Both methods lead to the same answer, \(\frac{120}{169}\).