Problem 18 - 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. \(\tan ^{-1}(-26)\)

Step-by-Step Solution

Answer

\(\tan^{-1}(-26)\) is approximately \(-87.79979^{\circ}\) to five decimal places.

Step 1: Understanding Inverse Tangent Function

The inverse tangent function, denoted as \( \tan^{-1}(x) \) or \( \arctan(x) \), is used to find an angle whose tangent is \( x \). This function is defined for all real numbers, so \( \tan^{-1}(-26) \) is defined.

Step 2: Using a Calculator

To find \( \tan^{-1}(-26) \), you need to use a scientific calculator. Input \( -26 \) and press the \( \tan^{-1} \) or \( \arctan \) function button. Make sure your calculator is set to degrees mode since trigonometric functions might be in degrees or radians.

Step 3: Interpreting the Result

After calculating \( \tan^{-1}(-26) \) in degrees, the calculator should provide a result. For many calculators, the result will be a negative angle, as tangent is a periodic function with a range of \( (-90^{\circ}, 90^{\circ}) \). For example, you might find a result close to \(-87.79979^{\circ}\).

Step 4: Rounding to Five Decimal Places

Ensure that the output from your calculator is rounded to five decimal places. For \( \tan^{-1}(-26) \), you should write it as \(-87.79979^{\circ}\) after rounding to five decimal places.

Key Concepts

Arctangent

The arctangent, often denoted as \( \tan^{-1}(x) \) or more commonly as \( \arctan(x) \), is an important concept in trigonometry. It helps us find the angle whose tangent value is \( x \). In simpler terms, if \( \tan(A) = x \), then \( A = \tan^{-1}(x) \). This inverse trigonometric function is crucial for determining angles when you know the tangent value. It's defined for all real numbers, which means that for any real number you put into \( \tan^{-1}(x) \), you will get back some angle. Here, it's significant to understand that \( \arctan(x) \) maps input values to outputs in the range \(-90^\circ\) to \(90^\circ\) (or \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) in radians). This range limitation means that the function won't give you angles outside of this span. When working with negative inputs, like \( -26 \), the result will naturally also be a negative angle, revealing the symmetry and periodic nature of the tangent function.

Angle Measurement

Angle measurement is a fundamental component in mathematics, especially in trigonometry. Angles can be expressed in two main units: degrees and radians. Degrees are perhaps the more familiar unit to many people, with a complete circle being \(360^\circ\). On the other hand, radians are another way to measure angles, where a full circle is \(2\pi\) radians. The choice between degrees and radians depends often on the context and the preference. Inverse trigonometric functions, like the arctangent, can output results in either unit, depending on the calculator or method you're using. For practical purposes like solving problems with calculators, you may want to ensure your device is set according to the unit required. For example, finding \( \tan^{-1}(-26) \) in degrees will give you an angle close to \(-87.79979^\circ\) because the tangent function has a specific behavior for large negative inputs that restricts the expected output within the first and fourth quadrants of a circle.

Scientific Calculator Usage

Using a scientific calculator efficiently can greatly simplify finding values for trigonometric functions like the arctangent. Here is a quick guide to help you navigate the process:

• First, make sure your calculator is in the right mode, degree or radian, based on what the question requires.
• Entering the value is straightforward: you need to find the \( \arctan \) or \( \tan^{-1} \) button on your calculator. This button might be one of the secondary functions, so it could require using a shift or second function key.
• Once you have entered the value, say \(-26\), simply press the \( \arctan \) button.
• Your calculator should give you a result, which you can round off to five decimal places if required. For \( \tan^{-1}(-26) \), this would be approximately \(-87.79979^\circ\).

Ensuring the correctness of the calculator setting and input can prevent errors, leading to accurate and efficient problem-solving.