91Ó°ÊÓ

The tangent function, often represented as \( \tan \), is one of the basic trigonometric functions. It relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the adjacent side. Mathematically, \[ \tan( \theta ) = \frac{ \text{opposite} }{ \text{adjacent} } \] In the context of the unit circle, \[ \tan( \theta ) = \frac{ \text{sin}( \theta ) }{ \text{cos}( \theta ) } \]. For example, \( \tan \big( -\frac{\text{π}}{4} \big) = -1 \) occurs because \( sin \big( -\frac{\text{π}}{4} \big) \) is equal to \( -\frac{\text{√2}}{2} \) and \( cos \big( -\frac{\text{π}}{4} \big) \) is equal to \( \frac{\text{√2}}{2} \). Therefore, \[ \tan \big( -\frac{\text{π}}{4} \big) = \frac{-\frac{\text{√2}}{2}}{\frac{\text{√2}}{2}} = -1 \].

The tangent function is periodic with a period of \( \text{Ï€} \), meaning that \( \tan( \theta ) = \tan( \theta + \text{Ï€} ) \). It is undefined for angles where cosine is zero, such as \( \frac{\text{Ï€}}{2} \) and \( \frac{3\text{Ï€}}{2} \).

In our given problem, we need to find the tangent of \( -\frac{\text{Ï€}}{4} \). We know from the properties of the tangent function that \( \tan( -\frac{\text{Ï€}}{4} ) = -1 \). Now we can use this result for further calculations.

Angle measurement in trigonometry is often expressed in radians rather than degrees. One full circle is \( 360° \) or \( 2\text{π} \) radians. Here’s how they relate:

• \( 0° = 0 \text{ rad} \)
• \( 90° = \frac{\text{Ï€}}{2} \text{ rad} \)
• \( 180° = \text{Ï€} \text{ rad} \)
• \( 270° = \frac{3\text{Ï€}}{2} \text{ rad} \)
• \( 360° = 2\text{Ï€} \text{ rad} \)

For instance, \( -\frac{\text{π}}{4} \) radians is equivalent to \( -45° \). In the problem, we are asked to evaluate the tangent at this angle, which we have established as \( -1 \).

When dealing with trigonometric functions, angles can be positive or negative, depending on the direction of rotation:

• Positive angles are measured counterclockwise.
• Negative angles are measured clockwise.

Knowing how to convert between degrees and radians is essential.
Use the conversion formulas:
\[ \text{Degrees} = \frac{180°}{\text{π}} \times \text{radians} \] \[ \text{Radians} = \frac{\text{π}}{180°} \times \text{degrees} \].
In the context of our exercise, understanding radians helps to determine the correct angle for inverse trigonometric functions.

Inverse trigonometric functions help us find the angle given the value of a trigonometric ratio. Some common inverse trigonometric functions are:

• \(\text{sin}^{-1}(x)\) or \(\text{arcsin}(x)\) for the inverse of the sine function.
• \(\text{cos}^{-1}(x)\) or \(\text{arccos}(x)\) for the inverse of the cosine function.
• \(\text{tan}^{-1}(x)\) or \(\text{arctan}(x)\) for the inverse of the tangent function.

In the exercise, we use the inverse cosine function \( \text{cos}^{-1}(x) \). It provides the angle whose cosine is given:

• \( \text{cos}^{-1}(1) = 0 \)
• \( \text{cos}^{-1}(0) = \frac{\text{Ï€}}{2} \)
• \( \text{cos}^{-1}(-1) = \text{Ï€} \)

The range of \( \text{cos}^{-1}(x) \) is [0, \( \text{Ï€} \)], which means it returns values within this range.

In our problem, we have \( \text{cos}^{-1}(-1) \). From the properties of inverse cosine, we know that the angle whose cosine is \(-1\) is \( \text{Ï€} \). Thus, the exact value of \( \text{cos}^{-1}[ \tan( -\frac{\text{Ï€}}{4} )] \) is \( \text{Ï€} \).
Inverse trigonometric functions are crucial for solving equations where angles must be determined from trigonometric ratios. They help us traverse back from the trigonometric values to the angles.