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Inverse trigonometric functions are essential in understanding relationships between angles and ratios of sides in right-angled triangles.
Specifically, the arctangent (or inverse tangent) function, \(\tan^{-1}(x)\), gives us the angle whose tangent value is \x\.
These functions are the inverse operations of the standard trigonometric functions (sine, cosine, and tangent) and are fundamental in solving for angles in various applications.
Typically, inverse trigonometric functions are denoted as:

• \(\tan^{-1}(x)\): arctangent
• \(\text{sin}^{-1}(x)\): arcsine
• \(\text{cos}^{-1}(x)\): arccosine

When solving our problem with \(\tan^{-1} (5.68)\), we find the angle whose tangent value is 5.68.

The tangent function, often abbreviated as tan, is one of the six primary trigonometric functions in mathematics.
To recall, it is defined in a right triangle as the ratio of the length of the opposite side to the length of the adjacent side of an angle.
Mathematically, it is expressed as: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \] The tangent function also has properties and graphs that show its behavior over different angles.
Importantly, the reciprocal of the tangent is the cotangent function.
Understanding the tangent is crucial when evaluating inverse trigonometric functions.
For instance, in calculating \(\tan^{-1}(5.68)\), we're seeking the angle \(\theta)\) such that \[ \tan(\theta) = 5.68 \] Knowing these principles helps us understand why we're using the inverse tangent function to find the angle.

Angles can be measured in different units but the two most common are degrees and radians.
In many practical applications, particularly in navigation and engineering, degrees are often used because they divide a circle into 360 parts.
One key point when using a calculator to find the arctangent is to ensure it's set to the correct mode, usually, degrees for problems like ours.
Remember:

• 360 degrees is a full circle
• 180 degrees is a straight line
• 90 degrees forms a right angle

The calculation of \(\tan^{-1}(5.68)\) in degrees means we find the angle in this unit.
Calculators often allow for easy switch modes between radians and degrees.

Using a calculator to find values of inverse trigonometric functions is straightforward if you follow a few steps:

• Ensure your calculator is set to degree mode if the problem requires an answer in degrees.
• Enter the number 5.68.
• Press the \( \tan^{-1} \text{ or arctan} \) button.
• Look at the display to find the result.

In our example, the calculator's function returns the angle in degrees whose tangent value is 5.68.
This is typically a time when you'll use functions marked as \( \tan^{-1} \) or something similar depending on your calculator's brand.
For instance:

• \(\tan^{-1} (5.68) = 80.567\)
• Round it to the nearest tenth to get 80.6 degrees

These steps make it easy to solve trigonometric problems with accuracy.