Problem 36 - Algebra 2

Question

Use the function \(y=200\) tan \(x\) on the interval \(0^{\circ} \leq x \leq 141^{\circ} .\) Complete each ordered pair. Round your answers to the nearest whole number. $$ \left(141^{\circ}, \mathbb{}\right) $$

Step-by-Step Solution

Answer

The completed ordered pair is (141 degrees, y), where y is the whole number obtained from computing 200 * tan(141 degrees) and rounding it to the nearest whole number.

Step 1: Convert the Angle

The given angle is 141 degrees. Most calculators operate in radian mode by default, hence it is important to ensure the calculator is set to degree mode before proceeding.

Step 2: Substitute the Angle into the Function

After ensuring the calculator is set to degree mode, you will substitute the angle into the function. Therefore, the equation y = 200*tan(x) becomes y = 200* tan(141 degrees).

Step 3: Compute the Value of y

Using the calculator, compute 200 * tan(141 degrees) to determine the y value.

Step 4: Round the Answer

The exercise asks that the answers be rounded to the nearest whole number. Therefore, take the result obtained from the previous step and round it to the nearest whole number.

Key Concepts

Tangent Function

The tangent function is a fundamental trigonometric function often denoted as \( \tan(x) \). It is the ratio of the sine of an angle to the cosine of that angle. This ratio makes it a crucial tool in mathematics, particularly in solving problems involving right angles and trigonometric identities. When dealing with the function \(y = 200 \tan(x)\), the value of \(y\) will vary based on the angle \(x\).

Here’s a quick breakdown of how the tangent function works:

• Positive and Negative Values: The tangent function can produce both positive and negative values depending on the quadrant of the angle \(x\). For example, in the interval from \(90^\circ\) to \(180^\circ\), where \(141^\circ\) falls, \(\tan(x)\) is negative.
• Undefined Values: The tangent becomes undefined at odd multiples of \(90^\circ\) (or \(\pi/2\) radians), as the cosine of these angles equals zero.

Understanding and applying the tangent function correctly is essential for calculating the variable \(y\) in many trigonometric equations.

Angle Conversion

Many trigonometric problems require you to convert angles from degrees to radians or ensure that calculations are done in the right mode, especially when using calculators.

• Degree Mode: Most trigonometry problems at a basic level involve angles measured in degrees. It is important to make sure your calculator is in degree mode since we are dealing with degrees in this example ( 0^{\circ} \leq x \leq 141^{\circ}).
• Converting Degrees to Radians: Though not needed here, converting degrees to radians is a helpful skill. The conversion formula is: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \).

Always check your calculator settings before performing trigonometric calculations involving angles to avoid errors.

Rounding Numbers

Rounding is a vital mathematical skill used daily to simplify numbers to a specified degree of precision. In this exercise, we have computed \(200 \times \tan(141^\circ)\), and the next step is to round this number.

Here are the key points about rounding:

• Nearest Whole Number: Rounding to the nearest whole number means adjusting a number to the closest integer. If you have 247.6, it rounds to 248, while 247.4 rounds to 247.
• Rounding Rules: Typically, if the digit after the decimal is 5 or greater, you round up, otherwise, you round down.
• Purpose of Rounding: Rounding helps to prepare data for interpretation or reporting, providing a simpler number that is easier to remember or use.

In our exercise, the computed \(y\) value from \(200 \times \tan(141^{\circ})\) should be rounded to the nearest whole number to complete the ordered pair effectively.