Problem 59 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of inverse functions to find the exact value of the expression, if possible. \(\arctan \left(\tan \frac{11 \pi}{6}\right)\)

Step-by-Step Solution

Answer

The exact value of \( \arctan \left(\tan \frac{11 \pi}{6}\right) \) is \( \frac{\pi}{6} \).

Step 1: Identify the Interval of the Inverse Tangent Function

Inverse trigonometric functions are defined within specific intervals. For the arctangent (or inverse tangent) function, its range is \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \). We need to ensure that the value we calculate falls within this range.

Step 2: Evaluate the Interior of the Function

We need to compute the exact value of \( \tan \frac{11 \pi}{6} \). The tangent function period is \( \pi \), therefore \( \tan \frac{11 \pi}{6} \) equals to \( \tan \frac{\pi}{6} \), which is \( \frac{ \sqrt{3} }{3} \).

Step 3: Apply the Inverse Tangent

Now that we have the exact value of \( \tan \frac{11 \pi}{6} \) and it falls within the range of the arctangent function, we can simply take \( \arctan \left(\tan \frac{11 \pi}{6}\right) \) as \( \arctan \left(\frac{\sqrt{3}}{3}\right) \). Using the properties of the arctangent function, we know that \( \arctan \left(\frac{\sqrt{3}}{3}\right) \) equals \( \frac{\pi}{6} \) radians.

Key Concepts

Arctangent Function

The arctangent function is the inverse of the tangent function. Its primary role is to find an angle whose tangent is a known value. In mathematical terms, if you know the value of \( \tan(\theta) \), the arctangent will give you the measure of \( \theta \) itself. The symbol for the arctangent function is \( \arctan \).

A key aspect of the arctangent function is its range, which is restricted between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) radians. This limitation ensures that each angle value is unique and that the inverse function is properly defined. When solving problems involving \( \arctan \), always ensure that the resulting angle lies within this range. If your calculated angle does not fall within these bounds, it can get grouped into its equivalent value within the range.

The arctangent function helps convert a tangent ratio back into an angle, making it a useful function for various applications in trigonometry and geometry.

Tangent Function

The tangent function relates an angle of a right triangle to the ratio of its opposite side to its adjacent side. Specifically, for an angle \( \theta \), the function is represented as \( \tan(\theta) \).

One vital characteristic of the tangent function is its periodicity. The tangent function has a period of \( \pi \) because \( \tan(\theta) = \tan(\theta + n\pi) \) for any integer \( n \). This means that if you add or subtract \( \pi \) from an angle, the tangent will repeat its values periodically. This property can reduce the complexity of many exercises by simplifying angles to their smallest equivalent within a single period.

Calculations involving the tangent function often require familiarity with the specific values it can take, such as \( \tan(\pi/6) = \frac{\sqrt{3}}{3} \), which is crucial in exercises that ask for the evaluation of tangent and its inverse.

Trigonometric Identities

Trigonometric identities are equations that relate different trigonometric functions to one another. These identities are immensely helpful in simplifying complex trigonometric expressions and solving equations. Some of the core identities include the Pythagorean identity, angle sum and difference identities, and periodicity identities.

In solving problems like \( \arctan(\tan(11\pi/6)) \), recognizing periodicity is crucial. This allows us to reduce \( \tan(11\pi/6) \) to an equivalent term \( \tan(\pi/6) \) because the period of \( \pi \) ensures \( \tan(11\pi/6) = \tan(11\pi/6 - \pi) = \tan(\pi/6) \).

Utilizing trigonometric identities, like the periodicity of the tangent function, is fundamental in breaking down and understanding the behavior of trig functions in various mathematical contexts.