The Tangent Function

1. Definition:

• The tangent function (tan(x)) is a trigonometric function that, in a right-angled triangle, is defined as the ratio of the length of the side opposite to the angle to the length of the side adjacent to the angle. 

• Formula:

  • tan(x) = Opposite / Adjacent

2. Graph of the Tangent Function:

• The graph of the tangent function has a distinctive shape:

  • It has vertical asymptotes at x = π/2, 3π/2, 5π/2, etc.

  • It repeats itself with a period of π.

  • It passes through the origin (0, 0).

3. Key Characteristics

• Domain: All real numbers except odd multiples of π/2 (x ≠ π/2 + nπ, where n is an integer).

• Range: All real numbers.

• Period: π

• Asymptotes: Vertical asymptotes at x = π/2 + nπ, where n is an integer.

4. Transformations

• General Form: y = A*tan(Bx - C) + D

  • A: Affects the amplitude (stretches or compresses the graph vertically).

  • B: Affects the period (changes the frequency of the graph).

  • C: Shifts the graph horizontally (phase shift).

  • D: Shifts the graph vertically.

5. Relationship to Sine and Cosine:

• The tangent function can be defined in terms of sine and cosine:

  • tan(x) = sin(x) / cos(x)

6. Applications

• The tangent function has various applications in fields like:

  • Physics: Calculating slopes, angles of elevation and depression, and analyzing wave phenomena.

  • Engineering: Designing structures, analyzing circuits, and solving problems in mechanics.

  • Navigation: Determining distances and directions.