Question

Find the exact value of each expression. Do not use a calculator. $$ \tan \frac{\pi}{4}+\cot \frac{\pi}{4} $$

Short answer

The exact value is 2.

Step-by-step solution

Recall the Value of \(\tan \frac{\pi}{4}\) and \(\cot \frac{\pi}{4}\)

The exact value of \(\tan \frac{\pi}{4}\) is 1 because the tangent of 45 degrees (or \(\frac{\pi}{4}\) radians) is 1. Similarly, the exact value of \(\cot \frac{\pi}{4}\) is 1 because the cotangent is the reciprocal of the tangent.

Add the Values

Sum the values of \(\tan \frac{\pi}{4}\) and \(\cot \frac{\pi}{4}\): \(\tan \frac{\pi}{4} + \cot \frac{\pi}{4}\). This is equal to \(1 + 1\).

Simplify the Expression

Simplify the expression to get the final result: \(1 + 1 = 2\).

Key concepts

tangent

The tangent function (\tan) is one of the primary trigonometric functions. It is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. More formally, for an angle \theta, the tangent function is expressed as:
ewline ewline \text{tan}(\theta) = \frac{\text{opposite}}{\text{adjacent}}
ewline ewline This function becomes useful in various fields such as physics, engineering, and computer graphics.
ewline ewline For angles like \frac{\text{\text{\(\theta\)}}{\text{\text{\(4\)}}}, the exact value can be determined easily without a calculator. For example, \text{tan}(\frac{\text{\text{\(\theta\)}}}{\text{\text{\(4\)}}}) = 1 because the tangent of 45 degrees (or \frac{\text{\text{\(\theta\))}}{4 radians)) is 1. This is a key value to remember when solving trigonometric problems.

cotangent

The cotangent function (\text{cot}) is another important trigonometric function. It is the reciprocal of the tangent function. That means, for an angle \theta, cotangent is given by:
ewline ewline \text{cot}(\theta) = \frac{1}{\text{tan}(\theta)} = \frac{\text{adjacent}}{\text{opposite}}
ewline ewline In practical terms, it’s often more convenient to use when you are given the lengths of the adjacent and opposite sides of a triangle.
ewline ewline For the specific angle of \frac{\text{\text{\(\theta\)}}}{\text{\text{\(4\)}}}, the cotangent value is also 1. This is because \text{cot}(\frac{\text{\text{\(\theta\)}}}{\text{\text{\(4\)}}}) = \frac{1}{\text{tan}(\frac{\text{\text{\(\theta\)}}}{\text{\text{\(4\)}}})} = 1. Understanding this reciprocal relationship is crucial for solving problems where both tangent and cotangent are involved.

exact value

When we talk about the 'exact value' of a trigonometric function, we refer to the precise value without any approximations. These values are typically derived from well-known angles like 0, 30, 45, 60, and 90 degrees, or their radian equivalents (\frac{\text{\(\theta\)}}{6}, \frac{\text{\(\theta\)}}{4}, etc.).
ewline ewline For example, finding the exact value of the sum of \text{tan}(\frac{\theta}{4}) + \text{cot}(\frac{\theta}{4}) involves knowing that both \text{tan}(\frac{\theta}{4}) and \text{cot}(\frac{\theta}{4}) are 1. Thus, the exact value is 1 + 1 = 2, without needing any decimal approximations.
ewline ewline Knowing the exact values of these trigonometric functions at common angles can significantly simplify solving more complex equations and is essential for a solid understanding of trigonometry.