AOS1 Topic 6: Circular Functions (TAN)

The tangent function is one of the primary circular or trigonometric functions, along with sine and cosine. It plays a crucial role in understanding the relationships between angles and side lengths in right triangles, as well as in the context of the unit circle.


Definition of Tangent


In a right triangle, the tangent of an angle \( \theta \) is defined as the ratio of the length of the opposite side to the length of the adjacent side:


\(\text{tan}(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)


This definition comes directly from the geometric interpretation of a triangle. However, in the unit circle (a circle with radius 1 centered at the origin), the tangent function can also be understood in terms of sine and cosine:


\(\text{tan}(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)


Since sine and cosine represent the y- and x-coordinates of points on the unit circle, respectively, the tangent function relates these coordinates.



Domain and Range of Tangent


The domain of the tangent function includes all real numbers except for odd multiples of \( \frac{\pi}{2} \) (90 degrees). At these points, the cosine of the angle is zero, and since the tangent function involves division by cosine, the function becomes undefined.


In interval notation, the domain of \(\text{tan}(\theta)\) is:


\(\theta \in \mathbb{R} \setminus \left\{ \frac{\pi}{2} + n\pi \mid n \in \mathbb{Z} \right\}\)


The range of the tangent function is:


Range of \(\text{tan}(\theta) = (-\infty, \infty)\)


This is because the tangent function can take any real value, unlike sine and cosine, which are limited to the range \([-1, 1]\).


Periodicity of Tangent


The tangent function is periodic with a period of \( \pi \) radians (or 180 degrees). This means that the function repeats its values every \( \pi \) radians. Mathematically, this is expressed as:


\(\text{tan}(\theta + \pi) = \text{tan}(\theta)\)


This periodic behavior reflects how the tangent function behaves as the angle increases around the unit circle.


Asymptotes of Tangent


As mentioned earlier, the tangent function becomes undefined at odd multiples of \( \frac{\pi}{2} \), where the cosine of the angle is zero. At these points, the graph of the tangent function has vertical asymptotes. These asymptotes occur at:


\(\theta = \frac{\pi}{2} + n\pi, n \in \mathbb{Z}\)


Graph of Tangent


The graph of the tangent function exhibits a repeating pattern due to its periodicity. It features:


Vertical asymptotes at odd multiples of \( \frac{\pi}{2} \),

Zeros at multiples of \( \pi \), where sine is zero,


Increasing behavior between asymptotes, where it moves from negative infinity to positive infinity.


The tangent function’s graph repeats this pattern every \( \pi \) radians.



Created with GeoGebra ® , by Rabyia Manzoor, Link


Applications of Tangent


Trigonometry and Geometry: The tangent function is essential in solving problems related to right triangles, particularly when the angle and one side of the triangle are known.

Physics: In physics, tangent is used in analyzing forces, especially in situations involving slopes and angles of elevation or depression.

Engineering and Architecture: In fields like engineering and architecture, tangent plays a role in analyzing structures, especially when dealing with inclined planes or ramps.

Navigation: Pilots and sailors use tangent in navigation to calculate angles and distances when navigating by bearings.


Inverse Tangent Function


The inverse of the tangent function is called the arctangent or \( \text{tan}^{-1}(x) \). It gives the angle whose tangent is \( x \):


\(\theta = \text{tan}^{-1}(x)\)


The range of the inverse tangent function is restricted to \((- \frac{\pi}{2}, \frac{\pi}{2})\), ensuring that it is a well-defined function with a unique value for every input.



Created with GeoGebra ® , by Chris Nord, Link

Example 1

Using Tangent in the Unit Circle
Find tan(π/4) using the unit circle.

Solution:
On the unit circle, the coordinates of the point corresponding to θ = π/4 are (√2/2, √2/2), where:
sin(π/4) = √2/2, cos(π/4) = √2/2
Since tan(θ) = sin(θ) / cos(θ), we have:
tan(π/4) = (√2/2) / (√2/2) = 1
Therefore, tan(π/4) = 1.

Example 2

Finding an Angle from the Tangent
If tan(θ) = 2, find θ using the inverse tangent function.

Solution:
To find θ, use the inverse tangent (arctangent) function:
θ = tan-1(2)
Using a calculator, we find:
θ ≈ 63.43°
Thus, θ ≈ 63.43°.

Example 3

Solving a Real-World Problem with Tangent
A ladder leans against a wall, making an angle of 60 degrees with the ground. If the bottom of the ladder is 5 meters away from the wall, how long is the ladder?

Solution:
The tangent of the angle is the ratio of the opposite side (the height of the ladder against the wall) to the adjacent side (the distance from the wall):
tan(60°) = height / 5
We know that tan(60°) = √3. Solving for the height:
√3 = height / 5 → height = 5√3 ≈ 8.66 meters
Thus, the ladder reaches approximately 8.66 meters up the wall.

Example 4

Tangent Asymptotes
Find the vertical asymptotes of the tangent function for θ in the interval -2π ≤ θ ≤ 2π.

Solution:
Solution: The vertical asymptotes of tan(θ) occur where cos(θ) = 0, which happens at odd multiples of π/2:
θ = π/2 + nπ, where n ∈ Z
For -2π ≤ θ ≤ 2π, the asymptotes occur at:
θ = π/2, 3π/2, -π/2, -3π/2
Thus, the vertical asymptotes are at θ = ±π/2, ±3π/2.

Exercise 1

Exercise 2

Exercise 3

Exercise 4

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