The tangent function is the quotient of the sine and cosine functions. Specifically, tan(x) = sin(x)/cos(x). This relationship gives rise to certain values of x for which the tangent function is undefined, leading to its distinct graph with vertical asymptotes. For a deeper understanding of how these functions interact, refer to the graphs of sine.

**Key Features of the Tangent Graph**

**1. Periodicity**

- The tangent function is periodic, meaning it repeats its values at regular intervals. Specifically, its period is π. This periodic nature is evident as the graph exhibits the same pattern every π units.

**2. Asymptotes**

- The tangent graph has vertical asymptotes, which are vertical lines the graph approaches but never touches. These asymptotes occur where the cosine function is zero, as dividing by zero is undefined in mathematics. Mathematically, these points are given by x = π/2 + nπ, where n is any integer.
- As the function approaches these asymptotes, its value tends to infinity or negative infinity, depending on the direction of approach. For more on the basic concepts of function behaviour, see domain and range basics.

**3. Intercepts**

- The x-intercepts, or the points where the graph crosses the x-axis, occur where the sine function is zero. These points are given by x = nπ, where n is any integer.
- Since tan(x) = sin(x)/cos(x), whenever sin(x) is zero, tan(x) will also be zero, leading to these intercepts. Understanding the introduction to radians can help in visualising these intercepts.

**4. Range**

- The range of the tangent function encompasses all real numbers. This means it can take on any value from negative infinity to positive infinity, making its range (-∞, ∞).

**5. Symmetry**

- The tangent function is odd, which means it has rotational symmetry about the origin. This can be verified by the fact that tan(-x) = -tan(x).

**Transformations of the Tangent Graph**

**1. Vertical Stretch or Compression**

- When the tangent function is multiplied by a coefficient, it undergoes a vertical transformation. If the absolute value of this coefficient is greater than 1, the graph stretches vertically. Conversely, if it's between 0 and 1, the graph compresses vertically.

**2. Horizontal Stretch or Compression**

- Modifying the input of the tangent function by a coefficient results in a horizontal transformation. If the absolute value of this coefficient is greater than 1, the graph compresses horizontally. If it's between 0 and 1, the graph stretches horizontally. For more on these types of transformations, visit basic transformations.

**3. Phase (Horizontal) Shift**

- Adding or subtracting a constant to the function's input shifts the graph horizontally. For instance, tan(x + c) shifts the graph c units to the left, while tan(x - c) shifts it c units to the right.

**4. Vertical Shift**

- Adding or subtracting a constant outside the function shifts the graph vertically. This transformation moves the entire graph up or down by the value of the constant.

**Practical Applications and Example Questions**

The tangent function, with its unique properties, finds applications in various fields, including physics, engineering, and computer science. For instance, it's used in calculating slopes, analysing waveforms, and even in certain algorithms in computer graphics. To see how tangent relates to specific problems, check out the equation of a tangent line.

**Example 1:** Given a tangent graph that intersects the x-axis at x = 3π and x = 7π, determine the equation of the graph.

**Answer**: The x-intercepts of the tangent graph are given by x = nπ. If the graph intersects the x-axis at x = 3π and x = 7π, it means the graph has been shifted. The mid-point of these intercepts is 5π, which is the point of symmetry. Thus, the graph is y = tan(x - 5π).

**Example 2:** A tangent graph has vertical asymptotes at x = 2 and x = 2 + π. Determine the equation of the tangent graph.

**Answer**: The standard tangent graph has vertical asymptotes at x = π/2 + nπ. If our graph has an asymptote at x = 2, it means it has been shifted 2 units to the right. The equation of this graph is y = tan(x - 2). For additional examples and detailed explanations, refer to the section on graphs of sine.

## FAQ

A phase shift will horizontally translate the tangent graph left or right. For instance, adding a phase shift of π/4 to the tangent function will shift the entire graph π/4 units to the left. This means that all the features of the graph, including the vertical asymptotes and x-intercepts, will also move by the same amount in the same direction.

The tangent function is widely used in various fields such as physics, engineering, and computer graphics. One common application is in trigonometry to find the slope of a line. In physics, it's used to calculate angles of incidence and reflection. In navigation, the tangent function helps in determining distances and directions. Additionally, in architecture and design, it's used to create curves and angles that are not possible with just the sine and cosine functions.

The sine and cosine functions both have a fixed amplitude of 1, meaning their maximum and minimum values are 1 and -1, respectively. The tangent function, on the other hand, does not have a fixed amplitude. Its values can range from negative infinity to positive infinity, as seen by its vertical asymptotes. This is because the tangent function is the ratio of sine to cosine, and as the cosine function approaches zero, the value of the tangent function approaches infinity.

The tangent function is periodic with a period of π because it's the ratio of sine to cosine. Both sine and cosine functions have a period of 2π. However, since tangent is undefined wherever cosine is zero (which happens at odd multiples of π/2), the tangent function completes one full cycle between two consecutive vertical asymptotes, which are π units apart. This makes the period of the tangent function π, causing it to repeat every π units.

The tangent function is defined as the ratio of the sine function to the cosine function. The vertical asymptotes of the tangent graph occur at points where the cosine function is zero, because division by zero is undefined in mathematics. Specifically, for the standard tangent function, the cosine function is zero at odd multiples of π/2. As a result, the tangent function has vertical asymptotes at these points, indicating that the function approaches positive or negative infinity.

## Practice Questions

The period of the standard tangent function, y = tan(x), is π. If the period of y = tan(kx) is π/3, then the relationship between the periods is given by π/k = π/3. Solving for k, we get k = 3. Therefore, the value of k is 3.

The standard tangent graph has vertical asymptotes at x = π/2 + nπ. If our graph has its first asymptote at x = -π/4, it means it has been shifted left by π/4 units. The equation of this graph is y = tan(x + π/4).

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.