Problem 28 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Given the inherent variability between different models of calculators, providing a generic answer here is impractical. Therefore, the student should compute the value on their own calculator to the nearest two decimal places.

Step 1: Setting up the Calculator

Make sure your calculator is set to radian mode since the inverse trigonometric functions yield results in radians. Also, ensure it's set to compute to at least two decimal places.

Step 2: Input the Value

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

Step 3: Compute and Round Off

Hit the calculate or equals button to compute the value, then round off your result to the nearest two decimal places.

Key Concepts

Tangent

The tangent function is a fundamental part of trigonometry, representing the ratio of the opposite side to the adjacent side in a right triangle. The tangent function can be extended to all real numbers using the unit circle.
The inverse of the tangent function, often written as \( \tan^{-1}(x) \) or "arctan", provides the angle whose tangent is the value \( x \). It's important to understand that the arctan function is specifically defined to be a one-to-one function, usually giving results in the range of \(-\frac{\pi}{2} \) to \( \frac{\pi}{2} \) radians.

• The tangent function is periodic, with a period of \( \pi \).
• Inverse tangent will only return values within its principal range, ensuring a unique angle output.

This lies at the heart of trigonometric identities and applications.

Radians

Radians are a unit of angular measure used in many areas of mathematics. They offer a more natural way of understanding angles compared to degrees.
One radian is the angle created when the arc length equals the radius of the circle. There are \( 2\pi \) radians in a complete circle, which is equivalent to 360 degrees. Thus, converting between radians and degrees involves the formula:
\[ \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \] \[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \]
In the context of inverse trigonometric functions, calculators often provide outputs in radians, making it crucial to understand their role.

Calculator Use

Using a calculator correctly is vital for solving trigonometric problems. Here are some steps to ensure accurate calculations, especially when dealing with inverse trigonometric functions:

• Ensure your calculator is in the correct mode: radian mode for trigonometric computations.
• Check the precision: set your calculator to display at least two decimal places, more if precision is needed for subsequent calculations.
• Accurate Input: Carefully input \( \tan^{-1}(-30) \) to avoid errors. This involves selecting the inverse tangent function (often labeled as "arctan" or "atan").

These settings are crucial as trigonometric results in radians can be significantly different from degrees.

Rounding Decimal Places

Rounding numbers is an essential skill, especially when dealing with decimal results from calculators. It simplifies the interpretation and reporting of the answers.
To round to two decimal places, consider only the first three decimal digits. If the third digit is 5 or more, increase the second digit by one.
Here's a simple example: suppose the calculator gives a result of 1.654. To round to two decimal places:

• Look at the third digit: 4, which is less than 5.
• Keep the second digit unchanged: 1.65 remains 1.65.
• If it were 1.657, it would round to 1.66.

This approach ensures answers are concise and consistently understood across mathematical solutions.