Problem 58 - Precalculus: A Concise Course

Question

Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) $$ \cot (-0.9) $$

Step-by-Step Solution

Answer

To find \(\cot(-0.9)\), make sure the calculator is in radian mode, input -0.9, then apply the cotangent function and round to four decimal places. The exact answer may differ depending on the specific calculator used, but it will be approximately equal to -1.2602.

Step 1: Verify calculator mode

Ensure that the calculator is set to radian mode. This is because the cotangent function is being evaluated for an angle given in radians, not degrees.

Step 2: Input value

Next, input the given value (-0.9) into the calculator, making sure to enter it as negative, as the sign is important in trigonometric functions.

Step 3: Apply cotangent function

Apply the cotangent function on the calculator to the value -0.9 that already inputted.

Step 4: Reading and rounding off the result

The calculator will give a result, this result should be rounded off to four decimal places. This might be done automatically depending on the calculator, or it may need to be done manually.

Key Concepts

Cotangent Function

The cotangent function is one of the fundamental trigonometric functions. It is abbreviated as \( \cot \) and is closely related to the tangent function. Mathematically, it is defined as the reciprocal of the tangent function, which means: \( \cot(\theta) = \frac{1}{\tan(\theta)} \).
The cotangent function has a periodic nature, meaning it repeats its values in regular intervals. In the case of \( \cot(\theta) \), the period is \( \pi \) radians. This periodicity is important when dealing with angles greater than \( \pi \) or less than \( -\pi \).
Understanding when and how the cotangent function becomes undefined is also vital. \( \cot(\theta) \) is undefined whenever the tangent function is zero, which occurs at \( \theta = n\pi \), where \( n \) is an integer. This is because you cannot divide by zero.

Using a Calculator

Calculators are handy tools for evaluating trigonometric functions like the cotangent function, especially when angles are not simple values. To use the calculator effectively, follow these simple steps:

• Ensure the calculator is set to the correct mode (degrees or radians).
• Input the angle or value required.
• Select the trigonometric function desired.

Different calculators might have varied user interfaces, but generally, there are buttons specifically for trigonometric functions. If the cotangent function doesn't have a dedicated button, you can compute it using the reciprocal of the tangent function by dividing 1 by the tangent of the value. This manual method works universally if the calculator lacks a direct cotangent option.

Radians

Radians are an alternative to degrees for measuring angles. They are often preferred in mathematics because they provide a natural way to express angles and simplify formulas.
One radian is defined as the angle formed when the length of the arc is equal to the radius of the circle. The complete circle is \( 2\pi \) radians, which is equivalent to 360 degrees.
When dealing with trigonometric functions on calculators, especially for tasks like evaluating \( \cot(-0.9) \), make sure to set the calculator to radian mode. This ensures accurate computation, as mixing radians and degrees will lead to incorrect results.

Rounding Decimals

Rounding decimals is crucial in giving answers that are both accurate and easy to interpret. While using a calculator, you might encounter long decimal numbers that need to be rounded off to a specific number of decimal places.
This exercise requires rounding the result to four decimal places. Here's how you can do it:

• Identify the fourth decimal place (for instance, in 0.12345, the fourth place is 4).
• Look at the number immediately to the right (here, it's 5).
• If this number is 5 or greater, round the fourth decimal place up by one. If less than 5, leave it as it is.

Proper rounding ensures the result is appropriately precise without unnecessary detail, fitting the parameters given for the problem.