Tangent Function
The tangent function is one of the basic trigonometric functions and is quite a helpful tool in trigonometry.
Here are a few values of the tangent function.
In this mini-lesson, we will explore the world of tangent function in math.
You will get to learn about the tangent formula, tangent meaning, range and domain of the tangent function, tan function graph, trigonometric ratios, trig identities, and other interesting facts around the topic.
As we scroll down, try your hand at solving a few interesting practice questions at the end of the page.
Lesson Plan
What Is a Tangent Function?
The tangent function is one of the basic trigonometric functions.
Tangent is defined as the ratio of the opposite side to the adjacent side of a specific angle of a right-angled triangle.
Let's look at the right-angled triangle \(ABC\) shown below.
\(AC\) is the hypotenuse.
The base \(AB\) is the extended portion of \(\angle A\).
Hence, we call it the "side adjacent to \(\angle A\)".
The perpendicular side \(BC\) faces \(\angle A\).
Hence, we call it the "side opposite to \(\angle A\)".
Hence, the tangent formula is:
| \(\tan{A}=\dfrac{\text{Side Opposite to } \angle A}{\text{Side Adjacent to }\angle A}=\dfrac{BC}{AB}\) |
Tangent Using a Unit Circle
Let's understand the tangent meaning using a unit circle.
Look at the unit circle shown below.
The vertical line ED which touches the circle at only one point; at the point of tangency, E.
The triangles \(\Delta ABC\) and \(\Delta ADE\) are similar.
This follows that,
\[\begin{align}\dfrac{ED}{AE}&=\dfrac{CB}{AC}\\ \dfrac{\tan{A}}{1}&=\dfrac{\sin{A}}{\cos{A}}\\ \tan{A}&=\dfrac{\sin{A}}{\cos{A}}\end{align}\]
Basic Properties of Tangent Function
Some of the basic properties of tangent function are:
- The values of \(\tan{x}\) is undefined where \(\cos{x}=0\).
- The tangent function has infinite vertical asymptotes.
- The tangent function is the ratio of sine function and cosine function, that is, \(\tan{x}=\dfrac{\sin{x}}{\cos{x}}\).
Though we can find the values of cos, sin, and tan using the calculator, we can also refer to charts that have some standard angles 0o, 30o, 45o, 60o, and 90o.
Trigonometric Identities
An equation becomes an identity when it is true for all the variables involved in it.
Now, we will discuss trig identities involving tangent function which are true for all angles involved in it.
| \(1+\tan^{2}{A}=\sec^{2}{A}\) |
| \(\tan{(2A)}=\dfrac{2\tan{A}}{1-\tan^{2}{A}}\) |
| \(\sin{(2A)}=\dfrac{2\tan{A}}{1+\tan^{2}{A}}\) |
Tan Function Graph
Look at the tan function graph shown below.
Did you notice that the pattern of curves is repeating after an interval of \(\pi\)?
Also, observe that the values of \(\tan{x}\) increases as \(x\) increases in \(\left(0,\dfrac{\pi}{2}\right)\).
The function has vertical asymptotes at \(x=\dfrac{\pi}{2}, \dfrac{3\pi}{2}, -\dfrac{\pi}{2}\).
Range and Domain of Tangent Function and Other Properties
We make the following observations:
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The \(\tan x\) functions is not defined for all odd multiples of \(\frac{\pi }{2}\). This is because the base is 0 for these positions. Thus, the domain of the \(\tan x\) function is \(\mathbb{R}\ - \left\{ {\left( {2n + 1} \right)\frac{\pi }{2}} \right\}\), where \(n\) belongs to the set of integers.
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The range of this function is \(\mathbb{R}\), as the \(y\)-variation in the graph is from negative infinity to positive infinity.
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The \(\tan x\) function has a cycle of \(\pi \), instead of \(2\pi \).
Domain of the \(\tan x\) function is \(\mathbb{R}\ - \left\{ {\left( {2n + 1} \right)\frac{\pi }{2}} \right\}\), where \(n\) belongs to the set of integers and the range of \(\tan x\) function is \(\mathbb{R}\).
The other properties of the tangent function are listed below:
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The tangent function is a periodic function and it has a period of \(\pi\).
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The tan function is an odd function because \(\tan{(-x)}=-\tan{x}\).
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The slope of a straight line is the tangent of the angle made by the line with the positive \(x\)-axis.
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The graph of \(\tan{x}\) is symmetric with respect to the origin.
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The \(x\) intercepts of \(\tan{x}\) is where \(\sin{x}\) takes the values zero, that is, \(x=n\pi, n \in \mathbb{N}\).
Solved Examples
| Example 1 |
Find the tan value of the following triangle for the given angle \(\theta\).
Solution
In this triangle, the longest side or the side opposite to the right angle is \(AC\) and hence, \(AC\) is the hypotenuse.
The side opposite to \(\theta\) is \(AB\) and hence, \(AB\) is the opposite side.
The side adjacent to \(\theta\) is \(BC\) and hence, \(BC\) is the adjacent side.
Now we will find \(\tan \theta\) using the tan formula:
\[ \begin{align}\tan\theta &= \dfrac{ \text{Opposite}}{ \text{Adjacent}}=\dfrac{AB}{BC}=\dfrac{3}{4}\\[0.2cm] \end{align}\]
| \(\therefore\) \(\tan\theta = \dfrac{3}{4}\) |
| Example 2 |
Find the exact length of the shadow cast by a 15 ft tree when the angle of elevation of the sun is 60º.
Solution
Let us assume that the length of the shadow of the tree is \(x\) ft.
Using the given information:
Applying tan to the given triangle
\[ \begin{align} \tan 60^\circ &= \dfrac{15}{x}\\[0.2cm] x&= \dfrac{15}{\tan 60^\circ}\\[0.2cm] x &= \dfrac{15}{\sqrt 3} \,\,\, \text{(Using chart)} \\[0.2cm] x&= \dfrac{15 \sqrt 3}{3}\\ &x=5 \sqrt 3 \end{align} \]
| \(\therefore\) The length of the shadow of the tree is \(5 \sqrt 3\) ft |
| Example 3 |
Jack is standing on the ground and looking at the top of the tower with an angle of elevation of \(60^{\circ}\).
If he is standing 15 feet away from the foot of the tower, can you determine the height of the tower?
Solution
Since the ratio involves the sides \(AB\) and \(BC\), we will use the trigonometric ratio \(\tan{60^{\circ}}\).
\[\begin{align}\tan{60^{\circ}}&=\dfrac{AB}{BC}\\\sqrt{3}&=\dfrac{AB}{15}\\AB&=15\sqrt{3}\end{align}\]
| \(\therefore\) The height of the tower is \(15\sqrt{3}\) feet. |
- Can you determine the nature of the tangent function in 4 quadrants of the XY-plane using graphical representation?
- Can you draw the graph of \(\cot{x}\) using the graph of \(\tan{x}\)?
Interactive Questions
Here are a few activities for you to practice.
Select/type your answer and click the "Check Answer" button to see the result.
Let's Summarize
We hope you enjoyed learning about the Tangent Function with the examples and practice questions. Now, you will be able to easily solve problems on the tangent function in trigonometry.
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FAQs on Tangent Function
1. How do you use the word tangent?
The word tangent is used to address a line that touches a circle at a unique point.
2. Why is Tan called tangent?
The expanded form of tan is tangent. Tan function is related to the tangent of a unit circle.
3. How do you find tangent?
We can find the tangent using the following two ratios:
- \(\tan{x}=\dfrac{\sin{x}}{\cos{x}}\)
- \(\tan{\theta}=\dfrac{\text{Side Opposite to } \angle \theta}{\text{Side Adjacent to }\angle \theta}\)
4. How do you know if a function is periodic?
If the graph of a function keeps repeating its patterns after a definite interval, we say that the function is periodic in nature.
5. How do you find the tangent of a triangle?
We can find the tangent of a triangle using the following two ratios:
- \(\tan{x}=\dfrac{\sin{x}}{\cos{x}}\)
- \(\tan{\theta}=\dfrac{\text{Side Opposite to } \angle \theta}{\text{Side Adjacent to }\angle\theta}\)
6. What is the tangent rule?
The tangent rule is given by the following two ratios:
- \(\tan{x}=\dfrac{\sin{x}}{\cos{x}}\)
- \(\tan{\theta}=\dfrac{\text{Side Opposite to } \angle\theta}{\text{Side Adjacent to }\angle\theta}\)
7. What is tangent function in trigonometry?
We define the tangent function or tangent ratio of an angle as the ratio of the length of the opposite side to the length of the adjacent side with respect to a particular angle in a right triangle.