Problem 35 - A Graphical Approach to Precalculus with Limits

Question

Give the degree measure of \(\theta,\) if it exists. Do not use a calculator. $$\theta=\arctan (-1)$$

Step-by-Step Solution

Answer

\(\theta = -45°\) (or \(\theta = -\frac{\pi}{4}\) radians)

Step 1: Understand Inverse Trigonometric Functions

The function \(\arctan(x)\) is the inverse of \(\tan(\theta)\), where \(\theta\) is the angle whose tangent is \(x\). It returns an angle \(\theta\). For \(\arctan(x)\), the range of the angles \(\theta\) is between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) (or between -90° and 90°).

Step 2: Identify Tangent Value

The problem asks for \(\theta = \arctan(-1)\). This means we need an angle \(\theta\) where the tangent of the angle is \(-1\).

Step 3: Recall Key Tangent Values

Recall that \(\tan(\theta) = -1\) at certain specific angles such as \(-45°\) and \(135°\).

Step 4: Consider \(\arctan\) Range

For \(\arctan(x)\), consider only angles between \(-90°\) and \(90°\). Therefore, of 135° and -45°, only -45° fits within this range.

Step 5: Convert Degrees to Radians

As angles in trigonometric functions are often in radians, we need to convert degrees to radians. Since there are \(180°\) in \(\pi\) radians, \(-45°\) is \(-\frac{\pi}{4}\) radians.

Key Concepts

Understanding Arctan

Arctan, also known as the inverse tangent function, is crucial when solving trigonometric equations. Unlike the regular tangent function that takes an angle and gives back its tangent (a value), the arctan reverses this process. It takes a tangent value and returns the corresponding angle. This means that if you have a tangent value of something like

• -1,
• 0,
• 1,

the arctan will provide the angle where the tangent equals that number, within a specific range. For \ ext\arctan(x), the range of the output angle \(\theta\) is between \(-\frac{\pi}{2}\) and \(+\frac{\pi}{2}\), or in simpler terms, between -90° and 90°. This is why when we solve for \(\theta = \arctan(-1)\), we are asked to find an angle within this domain.

Navigating Degree Measure

Degrees are a common way to measure angles, and they're very intuitive – sort of like using a clock. Angles ranging from 0 to 360 reflect one complete turn around a circle. When dealing with inverse trigonometric functions like arctan, it's essential to remember that you could be working with negative degrees. Angles like -45° are entirely valid and used to denote the same direction as 315°, just taken in a different route. It's crucial to match the degree measurement to the function's range of results.

Tangent Value Insights

Understanding tangent values is key to making the most out of arctan calculations. Specifically, for basic angles, there are common tangent values to memorize.

• \(\tan(45°) = 1\)
• \(\tan(0°) = 0\)
• \(\tan(-45°) = -1\)

These values tell us the slope of the line if you think of the angle as a slope from zero.So for the specific problem of \(\theta = \arctan(-1)\), it means the angle \(\theta\) is one where the slope is \(-1\), and within the prescribed range of the arctan function, that particular angle is \(-45°\). This precise limit means we choose the special angles that fit within the -90° to 90° domain.