Problem 32 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a calculator to evaluate each function. Round your answers to four decimal places. (Be sure the calculator is in the correct angle mode.) (a) \(\tan 18.5^{\circ}\) (b) \(\cot 71.5^{\circ}\)

Step-by-Step Solution

Answer

The approximate value for \( \tan(18.5^{\circ}) \) and \( \cot(71.5^{\circ}) \) will depend on the measurements from the calculator. However, the process highlighted in the step-by-step solution will guide on how these values should be found.

Step 1: Find the Tangent

To solve part (a), make sure the calculator is in degree mode. Calculate the tangent of 18.5 degrees by inputting \( \tan(18.5) \) into the calculator.

Step 2: Find the Cotangent

For part (b), still in degree mode, calculate the cotangent of 71.5 degrees. Since calculators usually don't have a cotangent function, you'll have to take the reciprocal of the tangent, like this: \( \frac{1}{\tan(71.5)} \).

Key Concepts

Understanding Tangent

The tangent function is a trigonometric function often represented as \( \tan(x) \). It's one of the primary trigonometric functions along with sine and cosine. The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. Mathematically, it's expressed as:\[\tan(x) = \frac{\text{opposite}}{\text{adjacent}}\]

• Tangent is periodic with a period of \(180^{\circ}\) or \(\pi\) radians.
• Its values range from negative infinity to positive infinity.

The tangent function is used extensively in trigonometry, calculus, and geometry to solve problems involving angles and distances. When calculating angles with tangent, ensure your calculator is set to the correct mode (degrees or radians) to get accurate results. In this context, since angles are given in degrees, you must verify your calculator is in degree mode before computing \( \tan(18.5^{\circ}) \).

Demystifying Cotangent

The cotangent function, denoted as \( \cot(x) \), is the reciprocal of the tangent function. It can be represented mathematically as:\[\cot(x) = \frac{1}{\tan(x)} = \frac{\text{adjacent}}{\text{opposite}}\]

• Similar to tangent, cotangent is periodic, with periods of \(180^{\circ}\) or \(\pi\) radians.
• Cotangent values also range from negative infinity to positive infinity, but their graphs differ due to the reciprocal function.

Calculators generally don't have a direct cotangent button, so we often calculate it using the formula \( \cot(x) = \frac{1}{\tan(x)} \). In problems like the one at hand, to find \( \cot(71.5^{\circ}) \), compute the tangent first and then take its reciprocal. Remember, like with tangent, ensure your calculator is set to degree mode when the angle is in degrees.

Navigating Degree Mode

When dealing with trigonometric functions, it's crucial to ensure that your calculator is set to the appropriate angle measurement mode: degrees or radians. The degree mode interprets angle measurements as degrees, typical in many courses and practical applications.

• One full circle is \(360^{\circ}\).
• Trigonometric functions will produce different values depending on whether degrees or radians are used.

Before performing any trigonometric calculations, confirm your calculator is in the correct mode. For instance, when calculating \( \tan(18.5^{\circ}) \) or \( \cot(71.5^{\circ}) \), you should set it to degree mode in order to match the angle units provided in the exercise. This ensures your solutions are accurate and reflect the intended angle measurement.