Problem 50 - Precalculus : Building Concepts and Connections

Question

Use a scientific calculator to evaluate the giren trigonometric functions to four decimal places. $$\tan (-1.5)$$

Step-by-Step Solution

Answer

After following the steps with a scientific calculator, \(\tan(-1.5)\) equals to \( -14.1014\). Be sure to double-check your input and your rounding.

Step 1: Recognize the Function

First identify the trigonometric function to be calculated, which, in this case, is the tangent function. We are to evaluate \(\tan (-1.5)\). The argument, \(x\), of this function is -1.5.

Step 2: Input the Value into Calculator

Next, input the value -1.5 into your scientific calculator for the function tangent. Make sure to input the negative sign before 1.5. This will provide you with an initial result.

Step 3: Round to Four Decimal Places

The third step is to round the initial result to four decimal places. The use of four decimal places implies that you are to round your answer to the number line where the thousandths place is the furthest right digit. If the next number to it is 5 or above, you round up; otherwise, you leave it as it is.

Key Concepts

Scientific Calculator Usage

Scientific calculators are indispensable tools when it comes to solving complex mathematical problems, especially trigonometric functions. To effectively use a scientific calculator, there are several key steps and functionalities to become familiar with:

• Mode Selection: Ensure your calculator is in the right mode, such as degree or radian, depending on the trigonometric question at hand. For this exercise, confirm whether \(-1.5\) denotes degrees or radians. Adjust the calculator's settings accordingly.
• Correct Input: When entering negative numbers, always use the negative sign key. This guarantees your calculator interprets the value accurately.
• Function Keys: Trigonometric functions such as sine, cosine, and tangent are typically accessed with dedicated keys. Locate the "TAN" key for evaluating tangent functions. Enter your argument, like \(-1.5\), correctly.
• Reading Results: Scientific calculators will provide results typically in decimal form. Be prepared to adjust the result based on the requirements of your math problem, such as rounding to a specific decimal place.

By mastering these basic techniques, you can enhance your efficiency and accuracy in solving trigonometric problems on a scientific calculator.

Rounding Decimals

Rounding decimals is a fundamental skill in mathematics that ensures results are presented with the desired precision. When the requirement is to round to four decimal places, here are the steps you should follow:

• Identify the Decimal Place: First, locate the fourth decimal place. For any number, this corresponds to the fourth digit after the decimal point.
• Analyzing the Fifth Place: Look at the digit immediately following the fourth decimal place, the fifth digit, to guide your rounding decision.
• Rounding Rule: If the fifth digit is 5 or greater, increase the fourth digit by one. For example, if the calculation yields 0.34567, round it to 0.3457. If the fifth digit is 4 or less, keep the fourth digit unchanged.

This approach ensures that your results are consistently accurate, which is especially important when documenting final answers or when further calculations depend on your initial result.

Tangent Function Evaluation

Evaluating the tangent function \(\tan(x)\) is a key component of trigonometry, useful in various fields from engineering to geography. Here's a step-by-step guide to calculating the tangent of an angle, like \(-1.5\):

• Understanding Tan(x): The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the adjacent side. It's represented as \(\tan(x) = \frac{\text{opposite}}{\text{adjacent}}\).
• Sign Considerations for Negative Angles: Since tangent is a periodic function, \(\tan(-x) = -\tan(x)\). This property helps when dealing with negative angles, meaning \(\tan(-1.5)\) is simply the negative of \(\tan(1.5)\).
• Calculator Steps: Use the calculator's tangent function, often labeled as "TAN," and input the angle. A scientific calculator will immediately compute this and display the result.

Understanding these principles helps in intuitively grasping how the tangent function behaves across different angle measures and enhances problem-solving skills in trigonometry.