Problem 63 - Precalculus : Building Concepts and Connections

Question

Evaluate the given expressions to four decimal places with a calculator. $$\cot ^{-1}(-3.2)$$

Step-by-Step Solution

Answer

The output will be the calculated angle in radians to four decimal places. The actual result could vary based on the calculator or programming language used, as some may use degrees instead of radians for trigonometric functions.

Step 1: Interpret the function

The function \( \cot^{-1}(-3.2) \) represents the angle whose cotangent is -3.2. This is the inverse cotangent function, also known as arccotangent.

Step 2: Use a calculator to compute

Use a scientific calculator with inverse trigonometrics functions. If the calculator does not have a dedicated function for the inverse cotangent, you can use the identity \( \cot^{-1}(x) = \pi/2 - \tan^{-1}(x) \). So, \( \cot^{-1}(-3.2) = \pi/2 - \tan^{-1}(-3.2) \). Compute the result to four decimal places.

Step 3: Interpret the result

The result will be an angle in radians. This angle represents the cotangent's slope -3.2.

Key Concepts

Arccotangent

Arccotangent, often denoted as \( \cot^{-1}(x) \), is the inverse function of the cotangent. It returns the angle whose cotangent is a specific value. If you think of trigonometric functions and their inverses like a lock and key, the arccotangent would be the key that helps you find the angle when you already know the cotangent value.

While the standard cotangent function involves taking the ratio of the adjacent side to the opposite side in a right triangle, arccotangent finds the angle given that ratio. This is especially useful in trigonometry and calculus when solving equations involving right triangles or circles.

Cotangent

Cotangent, abbreviated as \( \cot \), is one of the six main trigonometric functions. It is the reciprocal of tangent, meaning \( \cot \theta = \frac{1}{\tan \theta} = \frac{\text{adjacent}}{\text{opposite}} \). In a right triangle, cotangent relates the lengths of the sides.

Cotangent can be thought of as the slope of the line stemming from the angle, where it measures how steep the angle is in relation to the x-axis of a circle. Even when dealing with negative values, like \(-3.2\), cotangent helps in determining the corresponding angle direction and magnitude.

• Useful in geometry
• Common in physics and engineering problems

Radians

Radians are a way of measuring angles, providing an alternative to degrees. One complete revolution around a circle is \(2\pi\) radians, equivalently 360 degrees. This makes \( \pi \) radians equal to 180 degrees.

Using radians can simplify many mathematical formulas, especially those in calculus. For trigonometric functions, radians are often preferred because they allow for more straightforward integration and differentiation, making the math cleaner and often more intuitive.

• More natural for calculus
• Simplifies expressions

Scientific Calculator

Scientific calculators are powerful tools for solving complex mathematical problems, including trigonometric functions like arccotangent. They typically have a range of capabilities beyond basic arithmetic, such as logarithmic functions, exponentiation, and of course, trigonometric functions.

When calculating the arccotangent or other inverse functions, you might need to use identities if the calculator doesn't have a dedicated button. For instance, \( \cot^{-1}(x) = \frac{\pi}{2} - \tan^{-1}(x) \) can be used when a direct arccotangent function is unavailable. Remember to ensure that the calculator is set to the correct mode, either radians or degrees, based on what your problem requires.