Problem 22 - Algebra and Trigonometry

Question

\(11-22\) . Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. \(\tan ^{-1}(-0.25713)\)

Step-by-Step Solution

Answer

\(-0.25105\)

Step 1: Understand the Problem

We need to find the inverse tangent (arctan) of \(-0.25713\). This will give us an angle in radians. We will compute this using a calculator and round the result to five decimal places.

Step 2: Use the Calculator

Use a scientific calculator to calculate \( an^{-1}(-0.25713)\). Ensure the calculator is set to radian mode since the inverse trigonometric functions typically return radian values.

Step 3: Calculate and Round

Using the calculator, the approximate value of \( an^{-1}(-0.25713)\) is calculated. As displayed on the calculator, this value is approximately \(-0.25105\) when rounded to five decimal places.

Key Concepts

Understanding Arctan

Inverse trigonometric functions, like arctan, are used to determine an angle when the value of the trigonometric function is known. - Arctan is the inverse of the tangent function, often written as \( \tan^{-1}(x) \). It returns the angle whose tangent is \(x\).- If the tangent of an angle \(\theta\) is a known value, then the arctan of that value gives \(\theta\).For instance, if you know \( \tan(\theta) = -0.25713 \), then \( \text{arctan}(-0.25713) = \theta \).It's important to note:- Arctan returns angles typically in radians between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).- This range is chosen as it provides a unique result for each input value, effectively making the inverse function a "one-to-one" relation.

Understanding Radian Measure

Radians are a unit of angular measure used in many areas of mathematics. One radian is the angle formed when the arc length is equal to the radius of the circle. The full circle is \(2\pi\) radians.Key facts about radian measure:- A complete revolution around a circle is \(360^\circ\) or \(2\pi\) radians.- To convert degrees to radians, multiply by \(\frac{\pi}{180}\).- To convert radians to degrees, multiply by \(\frac{180}{\pi}\).Most trigonometric functions and their inverses will give output in radians because they are more natural in calculus and other advanced areas of math. For example, when you calculate \( \tan^{-1}(-0.25713) \), the result \(-0.25105\) is given in radians. Remember that using radian measure ensures the integration and derivation of trigonometric functions work seamlessly.

Using a Scientific Calculator

Calculating values for inverse trigonometric functions, like arctan, requires a scientific calculator. Here's how to effectively use one:1. **Setting the Mode:** - Before starting, set your calculator to "radian" mode because trigonometric calculations often default to radians. 2. **Calculating Arctan:** - Enter the function value, in this case, \(-0.25713\). - Use the inverse tangent button, often labeled as "tan\(^{-1}\)". This computes the arctan of the input. 3. **Rounding:** - After calculation, you'll obtain a value like \(-0.25105\). - Rounding to five decimal places ensures consistency and accuracy in your results.Using a calculator involves:- Checking your input and output settings, especially when switching between degrees and radians.- Verifying your results with manual approximations where applicable.