Understanding the Tangent Function and Its Key Properties: A Comprehensive Guide to Trigonometry’s Fundamental Ratio

Tangent

The tangent function, usually denoted as tan(x), is a trigonometric function that relates the length of the opposite side to the length of the adjacent side of a right triangle

The tangent function, usually denoted as tan(x), is a trigonometric function that relates the length of the opposite side to the length of the adjacent side of a right triangle. It is defined as the ratio of the sine of an angle to the cosine of the same angle.

The formula for tangent is:
tan(x) = sin(x) / cos(x)

Here are some key points to understand about the tangent function:

1. Range: The tangent function has a periodicity of π (pi), which means it repeats every π radians or 180 degrees. Its range is from negative infinity to positive infinity, which means it can produce any real number.

2. Periodicity: The tangent function repeats its values every π radians, or every 180 degrees. This means that tan(x + π) = tan(x), where x can be any real number.

3. Special Values: The tangent function has some special angles where its value is well-known. For example:
– tan(0) = 0
– tan(π/4) = 1
– tan(π/6) = √3 / 3
– tan(π/2) is undefined (as the cosine of π/2 is 0)

4. Symmetry: The tangent function is an odd function, which means that tan(-x) = -tan(x) for any value of x.

5. Graph: The graph of the tangent function has vertical asymptotes at x = (2n + 1)(π/2), where n is an integer. It also has points of discontinuity where the cosine of an angle is zero, as division by zero is undefined. The graph appears as a wave-like curve that oscillates between positive and negative infinity.

When working with tangent, you can use a calculator or trigonometric tables to find the tangent value of an angle. Keep in mind that the values may be rounded due to limitations on the accuracy of calculation.

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