Problem 14 - Precalculus Essentials

Question

Find the exact value of each expression. $$\tan ^{-1} 1$$

Step-by-Step Solution

Answer

The exact value of \( \tan^{-1}(1) \) is \( \frac{\pi}{4} \) radians or 45 degrees.

Step 1: Identify The Expression

The given expression is \( \tan^{-1}(1) \). This implies that we are finding the angle whose tangent is 1.

Step 2: Recall Special Angles

There are several angles associated with which the tangent gives a specific key values. In the case where \( \tan(\theta)=1 \), the corresponding angle is \( \frac{\pi}{4} \) or 45 degrees within the range from -90 to 90 degrees.

Step 3: Assign the Result

Therefore, the value of \( \tan^{-1}(1) \) is \( \frac{\pi}{4} \) or 45 degrees.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, also known as arc functions, provide us with an angle when we're given the trigonometric ratio. For instance, the arctangent, designated as \( \tan^{-1} \) or sometimes as \( \text{atan} \), offers a way to determine the angle when we know the tangent value.

These functions are essential for solving problems related to right-angled triangles and wave functions in physics. The arctangent, in particular, will yield an angle whose tangent is the number provided as its argument. What's more, the result returned is always within the principal branch for the function, which for \( \tan^{-1} \) is from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \) (or from -90 to 90 degrees). Unlike their trigonometric counterparts, which are periodic and have many angles that give the same ratio, the inverse functions ensure a unique value within their principal branch.

This uniqueness allows us to solve for unknown angles in geometrical problems, where previously we would only be able to state the ratio between the sides of a triangle. It is also widely used in calculus for finding the derivatives and integrals related to trigonometric functions.

Special Angles

Special angles in trigonometry refer to angles that produce exact and commonly recognized values for sine, cosine, and tangent functions. These angles are often used to simplify calculations and are also quite memorable.

Common special angles include \( 0^{\circ} \) (or \( 0 \) radians), \( 30^{\circ} \) (or \( \frac{\pi}{6} \) radians), \( 45^{\circ} \) (or \( \frac{\pi}{4} \) radians), \( 60^{\circ} \) (or \( \frac{\pi}{3} \) radians), and \( 90^{\circ} \) (or \( \frac{\pi}{2} \) radians). Recognizing these special values allows students to quickly identify the trigonometric function of these angles without having to use a calculator.

For example, the tangent of \( 45^{\circ} \) is 1 because the opposite and adjacent sides of the corresponding right triangle are of equal length. This also means that the tangent of \( \frac{\pi}{4} \) is 1. Familiarity with these angles can immensely speed up problem-solving and is key for understanding more complex trigonometric concepts.

Tangent Function

The tangent function is one of the six fundamental trigonometric functions. It is essentially the ratio of the opposite side to the adjacent side in a right-angled triangle, and it's denoted as \( \tan(\theta) \) where \( \theta \) is the angle of interest. Tangent is particularly special because it can also be understood as the ratio of the sine function to the cosine function, that is, \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).

Understanding the behavior of the tangent function is crucial for analyzing periodic phenomena, such as sound waves and alternating current signals. In practical applications, it's also used to calculate slopes and inclines. The function is periodic with period \( \pi \) radians, meaning it repeats its values every \( \pi \) radians along the angle axis. Unlike sine and cosine, the tangent function has asymptotes where the function is undefined, which occur at \( \frac{\pi}{2} + n\pi \) for any integer \( n \).

The exercise given demonstrates the use of the tangent function at a special angle; knowing that \( \tan(\frac{\pi}{4}) = 1 \) simplifies solving it without needing a calculator.