Question

Use the even-odd properties to find the exact value of each expression. Do not use a calculator. $$ \tan (-\pi) $$

Short answer

\(\tan(-\pi) = 0\)

Step-by-step solution

Understand the properties of the tangent function

Recall that the tangent function, \(\tan(x)\), is an odd function. This means that \(\tan(-x) = -\tan(x)\).

Apply the odd function property

To find \(\tan(-\pi)\), use the property for odd functions: \(\tan(-x) = -\tan(x)\). Therefore, \(\tan(-\pi) = -\tan(\pi)\).

Evaluate \(\tan(\pi)\)

Next, evaluate the tangent of \(\tan(\pi)\). Since \(\tan(\pi)\) is the tangent of a full period (\pi radians = 180 degrees), the value is 0. Thus, \(\tan(\pi) = 0\).

Find the final result

Using the result from step 3, substitute back into the equation from step 2: \(\tan(-\pi) = -\tan(\pi) = -0\). Hence, \(\tan(-\pi) = 0\).

Key concepts

Tangent Function

The tangent function, often written as \(\tan(x)\), is a fundamental trigonometric function that relates the angle of a right triangle to the ratios of its opposite side to its adjacent side. The tangent function is defined as follows:
\[ \tan(x) = \frac{\text{opposite}}{\text{adjacent}} \]
This means, for an angle x, if you know the lengths of the opposite and adjacent sides of the triangle, you can find the tangent of that angle by dividing the length of the opposite side by the length of the adjacent side.
Additionally, the tangent function has a period of \(\frac{\text{π}}{\text{2}}\), which means it repeats its values every \(\frac{\text{π}}{\text{2}}\) radians. Because of this periodicity, the tangent function also has certain properties that make it useful in solving not only geometric problems but also in applications involving wave and oscillatory phenomena.

Odd Functions

Odd functions have a unique property. For an odd function, if you take the input and negate it, the output is the negative of the output at the positive input. Mathematically:
\[ f(-x) = -f(x) \]
The tangent function \(\tan(x)\) is an example of an odd function. This means \(\tan(-x)\) will be equal to \- \tan(x) \.
So, for example, if you need to find \(\tan(-\text{π})\), using the property of odd functions, you would calculate it as follows:
\[ \tan(-\text{π}) = -\tan(\text{π}) \]
Using this property can simplify many trigonometric problems significantly. This is because you don't need to compute the tangent of a negative angle directly. You can compute the tangent of the positive angle and then just take the negative of that result.

Trigonometric Properties

Trigonometric functions have multiple properties that help in simplifying and solving problems. Some of the key properties are:

• Symmetry: Some functions are symmetric (even functions), while others are symmetric about the origin (odd functions).
• Periodicity: Functions like sine, cosine, and tangent repeat their values after a certain interval called the period. For tangent, this period is \(π\).
• Reciprocal Relationships: Each trigonometric function has a reciprocal function. For example, the reciprocal of the tangent function is the cotangent (\(\text{cot}(x) = \frac{1}{\tan(x)}\)).
• Pythagorean Identities: There are identities such as \( \tan^2(x) + 1 = \text{sec}^2(x)\), which relate different trigonometric functions to each other.

Using these properties in conjunction with each other can greatly simplify complex trigonometric expressions and make solving problems easier. They also provide a broader understanding of the behavior of these functions.

Tangent of π

The tangent function has interesting values at special angles. For the angle \(π\) (or 180 degrees), the tangent value is zero. This can be intuitively understood by considering the unit circle. At \(π\) radians, the point on the unit circle is \((-1, 0)\). In this position, the opposite side is zero (since the y-coordinate is zero), and therefore, the ratio of the opposite side to the adjacent side is also zero.
Hence, \[ \tan(\text{π}) = 0 \]
This becomes very useful when we have to find \( \tan(-\text{π}) \), as we can then use the odd function property to conclude that:
\[ \tan(-\text{π}) = -\tan(\text{π}) = -0 = 0 \]
Knowing these specific values can help solve more complex problems efficiently.