Question

Find the exact value of the expression. \(\tan \frac{17 \pi}{12}\)

Short answer

\(\tan(\frac{17 \pi}{12})\) equals to \(-2 - \sqrt{3}\)

Step-by-step solution

Break down the fraction

First, break down the fraction \(\frac{17 \pi}{12}\) into a sum of two fractions - \(\frac{\pi}{4}\) and \(\frac{\pi}{3}\), both of which corresponds to known angles in the unit circle. The sum of these two fractions gives us the original fraction: \(\frac{\pi}{4} + \(\frac{\pi}{3} = \(\frac{17 \pi}{12}\)

Apply the Tangent Sum formula

We can now use the Tangent Sum formula to find the tangent of the sum of two angles. The formula is: \(\tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}\). In this case, \(a = \frac{\pi}{4}\) and \(b = \frac{\pi}{3}\), so the formula becomes: \(\tan(\frac{17 \pi}{12}) = \frac{\tan(\frac{\pi}{4}) + \tan(\frac{\pi}{3})}{1 - \tan(\frac{\pi}{4})\tan(\frac{\pi}{3})}

Substitute tan values

\(\tan(\frac{\pi}{4}) = 1\) and \(\tan(\frac{\pi}{3}) = \sqrt{3}\). We substitute these values back into the formula to find the final result: \(\tan(\frac{17 \pi}{12}) = \frac{1 + \sqrt{3}}{1 - 1 * \sqrt{3}} = \frac{1 + \sqrt{3}}{1 - \sqrt{3}}\)

Simplify the result

To simplify this further, we can rationalize the denominator to get: \(\tan(\frac{17 \pi}{12}) = \frac{(1 + \sqrt{3})(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} = \frac{4 + 2\sqrt{3}}{1 - 3} = -2 - \sqrt{3}\)

Key concepts

Trigonometric Identities

At the heart of solving trigonometric problems is a deep understanding of trigonometric identities. These are equations that relate the trigonometric functions of angles to one another. They are used to simplify expressions, transform one type of function into another, and find the exact values of trigonometric functions for specific angles. Among the most crucial are the Pythagorean identities, double angle identities, and sum and difference formulas.

For example, the Tangent Sum formula, which is central to our exercise, falls under this category. It allows us to calculate the tangent of the sum of two angles if we know the tangent of each individual angle. The formula is expressed as \(\tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}\). Understanding and applying such identities correctly is essential for maneuvering through trigonometric problems with ease.

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of one unit, centered at the origin of a coordinate plane. The unit circle allows us to define the trigonometric functions for all angles, both positive and negative, and it extends beyond the introductory right triangle definitions that are limited to acute angles.

The angles and corresponding coordinates on the unit circle provide the exact trigonometric values for the sine, cosine, and tangent functions. For instance, at 45 degrees (\(\frac{\pi}{4}\) radians) and 60 degrees (\(\frac{\pi}{3}\) radians), the tangent values are 1 and \(\sqrt{3}\) respectively. Reference to the unit circle makes it possible to determine these values without memorization.

Rationalizing the Denominator

In mathematics, rationalizing the denominator is the process of eliminating the radicals (or complex numbers) from the denominator of a fraction. This process often involves multiplying the numerator and denominator by an appropriate form of 1 that will eliminate the radical without changing the value of the expression.

For example, if there is a single square root in the denominator, you would multiply by its conjugate. The conjugate is identical to the original expression but with the opposite sign between terms. In our exercise, we multiplied the original denominator \(1 - \sqrt{3}\) by its conjugate \(1 + \sqrt{3}\) to eliminate the square root. This step is not only important for simplification but also necessary for further algebraic manipulations and calculus operations.

Exact Trigonometric Values

Many trigonometric problems require finding the exact values of trigonometric functions for certain angles rather than approximate decimal values. These exact values are often expressed as fractions, square roots, or a combination of both. They are crucial for precise computations in mathematics, especially in geometry and calculus.

Some angles, like 30, 45, 60, and 90 degrees (or \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), and \(\frac{\pi}{2}\) in radians) have easily remembered exact trigonometric values, often found in reference to the unit circle or special triangles. Knowing these values by heart or knowing how to quickly derive them using the unit circle or other trigonometric identities is a significant skill for students solving trigonometry problems.