Domain and Range in Trigonometry: Simplified with Graphs

One of the key concepts in trigonometry is to understand the domain and range of both trigonometric functions and their inverses. These two sets of functions are closely related, and their properties help us solve various mathematical problems effectively.

Here is a quick guide for you as a reference during those exam days! Do pay attention to the observations and some simple ways to learn them.

1. Domain and Range of Trigonometric Functions

First thing to remember, in the case of trigonometric functions like sine (sin), cosine (cos), and tangent (tan):

○ The domain refers to the set of input values (angles) for which the function is defined,

○ While the range refers to the possible output values.

For instance: The domain of the sine and cosine functions is all real numbers, but their range is restricted to [-1, 1].

The tangent function (tan), however, has a domain of all real numbers except odd multiples of π/2, and its range is (-∞, ∞). The reason is simple, we know tan is defined as sin/cos, so whenever the denominator i.e. cos is equal to zero, tan will not be defined. And you know, cos is zero when the angle is an odd multiple of π/2.

Similarly, you can see for the other trigonometric functions.

Observations:

• Learn them pairwise as shown in different colours above.
• Moreover the remarks in blue colours in each case will help you to remember these values.

2. Domain and Range of Inverse Trigonometric Functions

On the other hand, inverse trigonometric functions like arcsin (sin⁻¹), arccos (cos⁻¹), and arctan (tan⁻¹) have these values that are essentially switched from their corresponding trigonometric functions. For instance:

○ The domain of the arcsin i.e. sin⁻¹ function is [-1, 1], and its range is [-π/2, π/2].

○ The domain of arccos i.e. cos⁻¹ is [-1, 1], and its range is [0, π].

○ Further, arctan (tan⁻¹) has a domain of (-∞, ∞) and a range of (-π/2, π/2).

Observations:

• As can be seen, the above table summarizes the domain and range of each function. The column on the left presents the domain of the trigonometric functions, while the right column correspondingly shows the ranges of their inverses in the respective Principal Value Branches.
• You’ll also notice that the range of the trigonometric function becomes the domain of its inverse. So, you can easily remember these values accordingly.

3. Graphs of All Functions

Additionally, the next set of images highlights the graphs of each pair.

In fact, these graphs also visually demonstrate how the functions and their inverses are related.

○ sin and sin⁻¹, cos and cos⁻¹

○ tan and tan⁻¹, cot and cot⁻¹

○ sec and sec⁻¹, cosec and coses⁻¹

Observations:

• You’ll see the graphs of trigonometric function and their corresponding inverses trigonometric functions are of the same shapes ofcourse. But, the two axes i.e. X and Y – axis got interchanged in each of the cases.
• Moreover, this interchange of the axes demonstrates the role reversal of domain and range (x and y) for each function with it’s inverse.
• You’ll also notice that in case of inverse functions, we are talking about Principal Value Branches — As a matter of fact, this basically directs us to discuss the invertibilty of trigonometric functions, that we shall do afterwards in the upcoming posts. (Do check the previous post on Trigonometric Identities)
• In the meantime, I suggest you to practice and draw all these graphs because that’s an effective way to learn them.

Practice Questions for You:

1. What is the range of the function cos⁻¹(x)?

2. Find the domain and range of the function sec(x).

3. Sketch the graph of the function arctan(x) and describe its domain and range.

4. Solve for x if sin⁻¹(1/2) = x.

You can try these. Moreover, I will keep adding and updating more important questions for practice in this section. Stay tuned by subscribing, so you don’t miss any of them!

Final Thoughts:

In conclusion, understanding the domain and range of trigonometric and inverse trigonometric functions is key to mastering these concepts. By using graphs and tables, you can visualize these relationships more easily. Keep practicing and exploring the graphs, you’ll gain a deeper understanding of how these functions behave. You can always refer to the images for a clear visual reference! And soon this knowledge will become second nature. Remember, math gets easier the more you engage with it! Happy learning!

---