Question

Evaluating inverse trigonometric functions Without using a calculator, evaluate or simplify the following expressions. $$\tan ^{-1}(\tan (\pi / 4))$$

Short answer

Question: Evaluate the expression $\tan^{-1}(\tan(\pi / 4))$ without using a calculator. Answer: $\pi/4$

Step-by-step solution

Recall the properties of inverse trigonometric functions

To begin solving this problem, we must understand the relationship between the tangent and arctangent functions. The arctangent function, denoted as \(\tan^{-1}(x)\) or \(\arctan(x)\), is the inverse of the tangent function. This means that for a given angle \(\theta\) in a right triangle, we have: $$\tan^{-1}(\tan(\theta)) = \theta$$ The purpose of inverse functions is essentially to "reverse" the effect of the original function. So given that our expression is in the form of \(\tan^{-1}(\tan(\pi / 4))\), we will use this property to simplify the expression.

Apply the inverse trigonometric property to the expression

Applying the inverse property of the tangent function to our given expression, we have: $$\tan^{-1}(\tan(\pi / 4)) = \pi / 4$$

Confirm the result is in the proper range

We must also make sure that our answer falls within the typical range of values for which the arctangent function is defined. The arctangent function produces results in the range \((-\pi / 2, \pi / 2)\). Since our answer, \(\pi / 4\), falls within this range, it is considered valid.

Final Answer

Therefore, the expression: $$\tan^{-1}(\tan(\pi / 4))$$ evaluates to: $$\pi / 4$$

Key concepts

Understanding Arctangent

The arctangent function, often written as \(\tan^{-1}(x)\) or \(\arctan(x)\), is the inverse of the tangent function. In simpler terms, it "undoes" what the tangent does. When you have an angle \(\theta\), the tangent tells you the ratio of the opposite side to the adjacent side in a right triangle.
Arctangent does the reverse: given a ratio, it finds the angle whose tangent is that ratio.
This function is incredibly useful in various fields, especially when angles need to be determined from ratios. One key thing to remember is that the arctangent will always give an output between \(-\pi/2\) and \(\pi/2\).

• Helps in finding the angle from a tangent value.
• Outputs are constrained to \((-\pi/2, \pi/2)\).

The Tangent Function

The tangent function, denoted as \(\tan(\theta)\), is a trigonometric function that relates to the angles and sides of a right triangle. Specifically, it represents the ratio of the length of the opposite side to the adjacent side of the angle \(\theta\).
This function follows a periodic pattern, repeating every \(\pi\), and is undefined at odd multiples of \(\pi/2\). It's crucial in solving problems involving angles and distances.

• Expresses the ratio of opposite to adjacent sides.
• Repeats every \(\pi\).

The tangent function plays an essential role in inverse trigonometric functions, as using \(\tan^{-1}\) helps retrieve the original angle from a ratio.

Exploring Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for any angle. One of the key identities used in inverse trigonometric functions is \(\tan^{-1}(\tan(\theta)) = \theta\), provided \(\theta\) is within the range of \((-\pi/2, \pi/2)\).
These identities help us simplify expressions and solve equations efficiently. They form the basis for understanding relationships between different trigonometric functions.

• Useful for simplifying expressions.
• Foundational for connecting trig functions.

Using these identities allows us to solve problems like evaluating \(\tan^{-1}(\tan(\pi/4))\), where we straightforwardly get \(\pi/4\) as the result, confirming the solution's validity.