Problem 28 - Precalculus Essentials

Question

Use a calculator to find the value of each expression rounded to two decimal places. $$\tan ^{-1}(-30)$$

Step-by-Step Solution

Answer

-1.57 radians or -89.97 degrees

Step 1: Understand the function

The symbolism \(\tan^{-1}\) represents the inverse tangent or arc tangent function, written as \(arctan\). Given an angle, the output of the inverse tangent function is the angle from the positive x-axis to the point (1, y) on the unit circle.

Step 2: Input into calculator

Input the following into your calculator: \(tan^{-1}(-30)\).

Step 3: Interpret the result

The result given by most calculators will be in radians because the default angle measurement in calculators is typically in radians rather than degrees. To convert this value to degrees multiply by \(\frac{180}{\pi}\).

Key Concepts

Arc Tangent

When you're working with trigonometric functions, it's important to understand their inverse counterparts. The inverse of the tangent function is known as the arc tangent, written as \( \tan^{-1} \) or \( \arctan \). This function helps us find angles when we know the tangent value. Specifically, \( \arctan(x) \) gives us an angle whose tangent is \( x \). So for \( \arctan(-30) \), we are trying to find an angle with a tangent of \(-30\).The arc tangent function returns an angle that corresponds to the point on a unit circle where the y-coordinate equals the given tangent value. These angles range between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) radians (or \(-90\) and \(90\) degrees). It's also worth noting that because the tangent function is periodic, its inverse doesn't give you all possible angles, but just a principal value within the mentioned range.

Radian to Degree Conversion

Angles can be measured in two main units: radians and degrees. While degrees are commonly used in day-to-day contexts, radians often crop up in mathematics because they relate directly to the concept of a circle's circumference. A radian measures the angle as a fraction of the circle’s total circumference.Most calculators will give you the result of a trigonometric function in radians. To convert this to degrees, you can use the formula:\[\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\]This means for any angle you get in radians, you can multiply it by \( \frac{180}{\pi} \) to switch to degrees. If your calculator returns \( \arctan(-30) \) in radians, remember this handy conversion to switch it to a more common measurement you might be more familiar with.

Calculator Usage

Using a calculator effectively is key to solving trigonometric problems. When dealing with functions like the arc tangent, automatic calculation saves time and reduces errors.Here's a simple guide to using your calculator for \( \arctan \):

• Check if your calculator is set to the right mode. For trigonometric calculations, this is usually radians by default.
• Input the function. You usually do this by pressing the \( \tan^{-1} \) button, followed by the number, such as \(-30\).
• Press the equals button to get your result.
• If needed, convert the result from radians to degrees using the radian-to-degree conversion formula.
• Round off the values as needed, often to two decimal points for precision.

These steps help ensure accurate results when calculating inverse trigonometric functions, like the arc tangent, using a calculator.