Problem 43 - Precalculus : Building Concepts and Connections

Question

Use a scientific calculator to evaluate the trigonometric functions. Make sure the calculator is in DEGREE mode. Round to four decimal places. $$\cot 67^{\circ}$$

Step-by-Step Solution

Answer

The cotangent of 67 degrees is approximately \(0.4049\), when rounded off to four decimal places.

Step 1: Set your calculator to DEGREE mode

For accurate results, make sure your scientific calculator is in degree mode. As the angle provided is in degrees, not having your calculator set to degree mode will give you a wrong answer.

Step 2: Input the angle

Once you ensure the calculator is in the right mode, input the angle 67 degrees into your calculator.

Step 3: Find the cotangent of 67 degrees

The next step is to find the cotangent of 67 degrees. Note that cotangent can be calculated as reciprocal of tangent function. So, compute the tangent of 67 degrees first and then take its reciprocal.

Step 4: Round to four decimal places

At last, round off the answer you get to four decimal places as directed in the problem.

Key Concepts

Using a Scientific Calculator

A scientific calculator is a powerful tool. It's designed to perform complex calculations, especially in math and science. To use a scientific calculator for trigonometric functions effectively, it's essential to familiarize yourself with its functions.

• Locate the trigonometric keys on your calculator, such as sine, cosine, and tangent.
• Ensure you know how to switch between different modes like degree and radian.
• Practicing these functions will make calculations quicker and help avoid errors.

When dealing with angles, using the calculator correctly is crucial to find accurate values of trigonometric functions like cotangent.

Understanding Degree Mode

Degree mode is used for calculations involving angles measured in degrees. Many calculators have multiple modes, so setting the correct one is vital.

• Check your calculator display to confirm it is in degree mode, often indicated as 'DEG'.
• Switching modes is usually done through a mode button or settings menu.

If you mistakenly calculate in radian mode instead of degree mode, you'll end up with incorrect answers. Remember, degrees are most commonly used in everyday problems, especially ones involving triangles.

Steps for Cotangent Calculation

Cotangent is a trigonometric function that is the reciprocal of the tangent function. To find \cot 67^\circ\, follow these simple steps:

• First, ensure your calculator is in degree mode.
• Input the angle, in this case, 67 degrees.
• Calculate the tangent: \(\tan 67^\circ\).
• Find the reciprocal of the tangent result: \(\cot 67^\circ = \frac{1}{\tan 67^\circ}\).
• Finally, round off your result to four decimal places for precision.

This procedure ensures accurate cotangent values when using a calculator.

Exploring the Tangent Function

The tangent function is a fundamental trigonometric function. It relates the angle of a right triangle to the ratio of the opposite side to the adjacent side. Here's how it connects with cotangent:

• \(\tan \theta\) gives the ratio of the opposite to the adjacent side.
• \(\cot \theta = \frac{1}{\tan \theta}\), meaning cotangent is the inverse of this ratio.

Knowing how to compute tangent and its reciprocal is key for solving many trigonometric problems. Especially for angles like 67 degrees, where direct cotangent keys may not be available, understanding this relationship is crucial.