Problem 32

Question

Find the exact value of the given trigonometric expression. Do not use a calculator. $$ \arctan \left(\tan \frac{\pi}{7}\right) $$

Step-by-Step Solution

Verified

Answer

The exact value is $\frac{\pi}{7}$.

1Step 1: Understanding the Relationship

The problem involves $\arctan$ and $\tan$, which are inverse functions. If $ -\frac{\pi}{2} < x < \frac{\pi}{2} \, \arctan(\tan(x)) = x$. We are given $\arctan\left(\tan\frac{\pi}{7}\right)$, and we need to verify if the angle $\frac{\pi}{7}$ is within the range of $\arctan$.

2Step 2: Checking the Range of Arctan

The given angle $\frac{\pi}{7}$ is approximately 0.4488 radians. Since this value lies between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$, it is within the range where $\arctan(\tan(x)) = x$.

3Step 3: Applying Inverse Function Property

Because $\frac{\pi}{7}$ is within the valid range, we can directly apply the inverse property. Therefore, $\arctan(\tan\frac{\pi}{7}) = \frac{\pi}{7}$.

4Step 4: Final Answer

The solution to the problem is that the exact value of $\arctan(\tan\frac{\pi}{7})$ is simply $\frac{\pi}{7}$.

Key Concepts

Understanding ArctanExploring the Tangent FunctionThe Angle Range of \$\arctan\$ FunctionRole of Precalculus in Understanding Inverse Functions

Understanding Arctan

The arctan, or inverse tangent function, is the opposite of the tangent function. Its primary role involves determining the angle, whose tangent value is given. Let's say you have a tangent of 1, the arctan of 1 is the angle that gives a tangent of 1. Within the range of $-\frac{\pi}{2}$ to $\frac{\pi}{2}$, arctan delivers clear results, making it a useful tool in trigonometry for finding angles. This function guarantees outputs that fit standard trigonometric circle constraints, ensuring that the results are consistent and manageable.

Exploring the Tangent Function

The tangent function, denoted as tan, is one of the principal trigonometric functions. It relates an angle of a right triangle to the ratio of the opposite side over the adjacent side. Tangent functions are cyclical and repeat every $\pi$ radians, which is around 180 degrees. Knowing that the tangents repeat allows us to predict and understand their behaviors over a full cycle. The main property to remember is that tangent is periodic and can take any real value. Thus, while its inverse (arctan) contains certain limitations, tangent itself does not, covering an extensive set of real numbers.

The Angle Range of \$\arctan\$ Function

Understanding the angle range is crucial when dealing with inverse trigonometric functions like arctan. Here we need to focus on the range for $\arctan$ which is $-\frac{\pi}{2}$ to $\frac{\pi}{2}$. This range means that any angle calculated using arctan will fall within these boundaries. Because arctan and tan are inverse functions, the input to arctan should ideally produce outputs within this range to maintain accuracy and validity of results. In our specific exercise with $\arctan(\tan\frac{\pi}{7})$, it is confirmed valid since $\frac{\pi}{7}$ lies within the appropriate interval.

Role of Precalculus in Understanding Inverse Functions

Precalculus is an essential foundation that includes all the basic functions and sets the ground for calculus. Within precalculus, learning about inverses of trigonometric functions, like arctan and others such as arcsin and arccos, plays a pivotal role. It teaches students how to reverse these processes, turning results back into angles. This understanding is necessary for tackling complex calculus problems later. Furthermore, mastering precalculus concepts enables students to appreciate the mathematical relationships and the intricate balance between functions and their inverses, especially in contexts like determining exact values without computational aid.

Problem 32