Problem 48 - Algebra and Trigonometry

Question

\(47-50\) Find the exact value of the given expression. $$ \cos \left(2 \tan ^{-1} \frac{12}{5}\right) $$

Step-by-Step Solution

Answer

The exact value is \(-\frac{119}{169}\).

Step 1: Identify the Inverse Tangent

The expression involves the inverse tangent, \( \tan^{-1} \frac{12}{5} \). Let's denote \(. Suppose \theta = \tan^{-1} \frac{12}{5}\). This means that \( \tan \theta = \frac{12}{5} \).

Step 2: Use the Double Angle Formula for Cosine

The next step uses the double angle identity for cosine: \( \cos(2\theta) = 1 - 2\sin^2(\theta) \) or \(\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)\). Let's use \(\cos(2\theta) = 1 - 2\sin^2(\theta)\).

Step 3: Calculate Sin and Cos Using a Right Triangle

Since \( \tan \theta = \frac{12}{5} \), create a right triangle where the opposite side is 12, and the adjacent side is 5. The hypotenuse \( h \) can be found using the Pythagorean Theorem: \( h = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13\).

Step 4: Determine Values of Sin and Cos

From the triangle, \( \sin \theta = \frac{12}{13} \) and \( \cos \theta = \frac{5}{13} \).

Step 5: Apply Values to the Double Angle Formula

Substitute the values found into the double angle formula: \( \cos(2\theta) = 1 - 2\left(\frac{12}{13}\right)^2 \).

Step 6: Simplify the Expression

Compute \( \left(\frac{12}{13}\right)^2 = \frac{144}{169} \). Then multiply by 2: \( 2 \times \frac{144}{169} = \frac{288}{169}\).

Step 7: Finalize the Expression

Thus, \( \cos(2\theta) = 1 - \frac{288}{169} = \frac{169}{169} - \frac{288}{169} = \frac{-119}{169}\).

Key Concepts

Cosine Double Angle Formula

The cosine double angle formula is a trigonometric identity that can simplify expressions involving angles. It's particularly useful in this problem, where we're dealing with a value of cosine for an angle that is twice another angle. There are three variations of this formula:

• \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \)
• \( \cos(2\theta) = 2\cos^2(\theta) - 1 \)
• \( \cos(2\theta) = 1 - 2\sin^2(\theta) \)

In the solution, the third variation is used. The usage of these identities depends on the values you have or wish to find. These identities help by reducing the complexity of dealing with an angle like \(2\theta\). By expressing \(\cos(2\theta)\) in terms of \(\sin(\theta)\) and \(\cos(\theta)\), you can use known values to find expressions involving unknown compound angles.

Inverse Tangent

The inverse tangent function, also known as arctan, is used to find an angle whose tangent is a given number. In mathematical notation, if \( y = \tan^{-1}(x) \), then \( \tan(y) = x \). This is a powerful tool in trigonometry because it allows us to determine angles based on the ratio of two sides of a right triangle.

For example, in the given problem, \( \theta = \tan^{-1}\left(\frac{12}{5}\right) \). Here, \( x = 12/5 \) represents the opposite-over-adjacent ratio in a right triangle. By drawing or visualizing such a triangle, it becomes easier to understand how this function helps us determine angles from ratios.

• Make a right triangle where the opposite side is 12 and the adjacent side is 5.
• Then use a trigonometric identity or a calculator to find the exact angle \( \theta \).

Understanding inverse functions like \( \tan^{-1} \) is crucial because they let us move between different mathematical representations, such as from side lengths to angles.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. The formula is \( a^2 + b^2 = c^2 \), where \( a \) and \( b \) are the legs of the triangle, and \( c \) is the hypotenuse, or the side opposite the right angle.

In this problem, the theorem is used to find the hypotenuse of a right triangle with sides 12 and 5:

• Calculate \( h = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \)

This step is crucial because knowing the variables \( \sin(\theta) \) and \( \cos(\theta) \) allows you to apply trigonometric identities more effectively. By finding all three sides of our triangle, we're able to compute both sine and cosine for \( \theta \), an essential step before applying the cosine double angle formula. Understanding and applying the Pythagorean theorem enables you to solve not only this specific math problem but also countless others involving right triangles.