Problem 12 - Algebra 2

Question

Solve each equation by finding the value of \(x\) to the nearest degree. \(\operatorname{Arctan} 1=x\)

Step-by-Step Solution

Answer

The value of \(x\) is 45 degrees.

Step 1: Understanding Arctan

The function \(\operatorname{Arctan}\) is the inverse of the tangent function. It returns the angle whose tangent is a given number. Hence, \(\operatorname{Arctan}(1)\) asks for the angle \(x\) whose tangent is 1.

Step 2: Applying the Property of Tangent

Remember that \(\tan(\frac{\pi}{4}) = 1\). This helps because \(\operatorname{Arctan}(1)\) corresponds to the angle \(\frac{\pi}{4}\) radians.

Step 3: Converting Radians to Degrees

To convert radians to degrees, use the formula: degrees = radians \(\times \frac{180}{\pi}\). For \(\frac{\pi}{4}\), this becomes \(\frac{\pi}{4} \times \frac{180}{\pi} = 45\) degrees.

Step 4: Final Result

Identify that the nearest integer degree for the angle whose tangent is 1 is 45 degrees, as calculated in the previous step.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are incredibly useful in math, especially when you need to determine angles from known trigonometric ratios. In this exercise, we are dealing specifically with the inverse tangent function, also known as \( ext{Arctan}\).

• This function helps you find the angle whose tangent value is given. Here, the arctan function provides the angle when the tangent equals a specified number.
• So, when you see \( ext{Arctan}(1) = x\), it is asking for an angle, let's call it \(x\), where the tangent of \(x\) equals 1.
• The inverse trigonometric functions only return principal values. For \( ext{Arctan}\), this restriction is between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) radians, ensuring the function is one-to-one and delivers a unique angle.

The principal value of \( ext{Arctan}(1)\) is \(\frac{\pi}{4}\) radians. This relationship is essential for solving equations that involve inverse trigonometric functions.

Angle Conversion

Angles can be represented in two main units: degrees and radians. Often, problems require converting between these units.

• In the given problem, we identified that the angle with a tangent of 1 was \(\frac{\pi}{4}\) radians.
• To convert radians to degrees, a straightforward formula is used: degrees = radians \(\times \frac{180}{\pi}\).
• For example, \(\frac{\pi}{4} \) radians multiplied by \(\frac{180}{\pi}\) yields \(45\) degrees.

Through this conversion, understanding becomes much simpler, as most are familiar with the degree system. It’s crucial to know such conversions for any tasks involving trigonometry or geometry.

Tangent Function

The tangent function is a fundamental trigonometric function that relates an angle of a right triangle to the ratios of its opposite and adjacent sides.

• It's defined as \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\).
• In the unit circle context, \(\tan(\theta)\) equals the \(y/x\) coordinate values of the angle on the circle.
• One of the key properties we utilized here is that \(\tan(\frac{\pi}{4}) = 1\). This is because, at \(45\) degrees or \(\frac{\pi}{4}\) radians, the opposite and adjacent sides of a triangle are equal.

Understanding the tangent function's behavior helps with so many trigonometry problems, especially when coupling it with its inverse function to solve equations.