Problem 57 - Precalculus : Building Concepts and Connections

Question

Use a scientific calculator to evaluate the giren trigonometric functions to four decimal places. $$\tan \left(-38^{\circ}\right)$$

Step-by-Step Solution

Answer

The tangent of -38 degrees rounded to four decimal places is \[-0.7813\].

Step 1: Calculating the Tangent

Firstly, to find the result of the tangent function, set your calculator to degree mode because the provided angle is in degrees. Then enter -38 into your calculator and apply the tangent function.

Step 2: Rounding Result

The calculator will show a result that may exceed four decimal places. Round this result to the nearest four decimal places. This can be done by looking at the fifth digit; if it's 5 or larger, increase the fourth digit by 1, else leave it as it is.

Step 3: Final Answer

The result after rounding to four decimal places is the final answer.

Key Concepts

Scientific Calculator

Scientific calculators are essential tools for solving trigonometric functions, such as the tangent function. These calculators can handle a wide range of mathematical operations and are designed to work with functions involving angles, such as sine, cosine, and tangent. To calculate the tangent of an angle using a scientific calculator, ensure it is in the correct mode. Since angles can be measured in degrees or radians, make sure your calculator is set to 'degree' mode when working with degrees. This setting is usually indicated as 'DEG' on the calculator display.

To evaluate a function like \( \tan(-38^{\circ}) \), enter the angle into your calculator by pressing the '+' or '−' button as necessary, followed by the 'TAN' button. The calculator will then compute the tangent of that angle, providing you with the unrounded result.

Using a scientific calculator correctly will make solving trigonometric problems much more straightforward, allowing you to focus on the calculations rather than the computation process itself. This makes them indispensable in both educational settings and various professional fields.

Tangent Function

The tangent function is a fundamental trigonometric function represented as \( \tan \), and it relates an angle of a right triangle to the ratio of the length of the opposite side to the adjacent side. In formula terms, \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).

One unique aspect of the tangent function is its periodic nature; it repeats its values over a cycle of \( 180^{\circ} \) or \( \pi \) radians. The tangent function is also defined for all real numbers, except where the cosine function (the denominator) equals zero. This occurs at angles like \( 90^{\circ}, 270^{\circ}, \) etc., where tangent is undefined.

It's important to remember that trigonometric functions can have both positive and negative values depending on the quadrant in which the angle lies. Negative angles are measured clockwise from the positive x-axis, so \( -38^{\circ} \) falls in the fourth quadrant where the tangent is negative. Understanding these nuances is critical in accurately using and interpreting trigonometric functions.

Rounding Decimal Places

Rounding is an essential mathematical process to make numbers easier to work with, especially when dealing with decimals. When asked to round a result to four decimal places using a calculator, first observe the digit in the fifth position. This digit determines whether the fourth decimal place remains as it is or increments by one.

Here’s a simple rule:

• If the fifth decimal digit is 5 or greater, increase the fourth decimal digit by one.
• If the fifth decimal digit is less than 5, leave the fourth decimal digit unchanged.

For example, if your calculator displays \(-0.781285\), you would look at the fifth decimal digit, which is '8' in this case. Since '8' is greater than '5', you round up the fourth decimal place, changing \(-0.7812\) to \(-0.7813\).

Rounding is crucial for precision, especially when the results need to meet specific requirements such as homework, scientific calculations, or even professional documentation. Proper rounding ensures consistency and accuracy in your answers.