Problem 21 - Contemporary Precalculus

Question

Use a calculator in radian mode to approximate the functional value. $$\tan ^{-1}[\tan (-4)]$$

Step-by-Step Solution

Answer

Question: Find the value of the inverse tangent function of the tangent function evaluated at -4 radians using a calculator in radian mode. Answer: To find the value of the inverse tangent function of the tangent function evaluated at -4 radians, follow these steps: 1. Ensure your calculator is in radian mode. 2. Calculate the value of the tangent function at -4 radians: $$\tan(-4)$$. 3. Evaluate the inverse tangent function using the calculated tangent value: $$\tan^{-1}[\tan(-4)]$$. 4. Verify if the result is within the range of the inverse tangent function, between $$-\frac{\pi}{2}$$ and $$\frac{\pi}{2}$$, and adjust if necessary. Note: Converting radians to degrees is optional but can help visualize the angle.

Step 1: Convert radians to degree (optional)

To understand better the value, we can convert -4 radians to degrees:$$-4 \times \frac{180}{\pi} \approx -229.18°$$This step is optional but helps visualize the angle in degrees. Remember to use the calculator in radian mode for the following steps.

Step 2: Find the tangent value of the input

Calculate the value of the tangent function at -4 radians:$$\tan(-4)$$Use the calculator in radian mode to evaluate the tangent value.

Step 3: Evaluate the inverse tangent function

Now, evaluate the inverse tangent function using the calculated tangent value from previous step:$$\tan^{-1}[\tan(-4)]$$Use the calculator to find the approximate value of the inverse tangent function.

Step 4: Verify the output's range

Finally, verify if the result is within the range of the inverse tangent function. The output range of the arctangent function is between $$-\frac{\pi}{2}$$ and $$\frac{\pi}{2}$$. If necessary, adjust the result so that it lies within this interval.

Key Concepts

Radian Mode Calculation

In mathematics, angles can be represented in degrees or radians, the latter being the standard mode in higher mathematics and many scientific fields. The radian is the standard unit of angular measure used in many areas of mathematics. For any circle, the length of an arc of the circle is directly proportional to the radian measure of the angle subtended by that arc. One full revolution of a circle corresponds to an angle of \(2\pi\) radians which is equivalent to 360 degrees.

When using a calculator to compute trigonometric functions, it's crucial to ensure that the calculator is set to the correct mode in accordance to the unit of measure used for the angle, whether degrees or radians. This is especially important because the values of the trigonometric functions differ vastly for the same numerical value if the units are not consistent. The radian mode calculation involves the calculator interpreting the input angle as radians. In the exercise provided, you would set your calculator to radian mode to approximate the functional value of an inverse tangent of a tangent.

Tangent and Inverse Tangent

The tangent function, often abbreviated as \(\tan\) is one of the three primary trigonometric functions. It relates the angle of a right triangle to the ratio of the opposite side to the adjacent side. Tangent is particularly useful in situations involving slopes, angles of elevation or depression, and in solving triangles.

The inverse of the tangent function is the arctangent, denoted by \(\tan^{-1}\) or sometimes \(\arctan\). The arctangent is used to find an angle whose tangent is a given number. It effectively 'reverses' the action of the tangent — where given the ratio of the opposite to the adjacent side, one can determine the angle. However, due to the periodicity of the tangent function, its inverse does not uniquely determine an angle without additional constraints. The range of the arctangent function is from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \), reflecting the fact that the tangent function increases without bound as the angle approaches either \( \frac{\pi}{2} \) or \( -\frac{\pi}{2} \) radians. Thus, when using the inverse tangent, the result must fall within this interval, which includes the understanding that the output of \(\tan^{-1}(\tan(\theta))\) gives the angle \(\theta\) that resides within this bound.

Converting Radians to Degrees

While calculations and formulas often use radians for measuring angles, our everyday language and conceptualization of angles frequently employ degrees. Conversion between these two units is, therefore, a quite common task in trigonometry. To convert an angle from radians to degrees, one would use the conversion factor \(\frac{180}{\pi}\).

The formula to convert an angle \(\theta\) from radians to degrees is: \[ \theta_{\text{degrees}}= \theta_{\text{radians}} \times \frac{180}{\pi} \. \] Conversely, to convert degrees to radians, multiply the angle in degrees by \(\frac{\pi}{180}\). It's important to make this conversion accurately to interpret the angle visually or when communicating with others who may be more familiar with degrees. In our exercise, converting -4 radians to degrees aids in visualization, but the calculator computations must remain in radian mode for precision and because trigonometric functions in calculators are based on radian measures.