Inverse Trigonometric Functions Class 12 | PlanetSpark

Trigonometry becomes more interesting in Class 12 when students are introduced to Inverse trigonometric functions. Until Class 11, we mostly worked with sine, cosine, and tangent. However, in Class 12, we start asking a different type of question. Instead of finding the value of sin 30 degrees, we try to find the angle whose sine is one by two.

That is where inverse functions come into the picture.

In this detailed guide, we will understand what is inverse trig functions, explore formulas, look at the inverse trigonometric functions table, learn properties, solve examples, and understand their importance in board exams.

What Are Inverse Trigonometric Functions

Let us first understand clearly what is inverse trig functions.

An inverse trigonometric function helps us find the angle when the value of a trigonometric ratio is given.

For example:

• sin 30 degrees equals one by two

• Therefore, sin inverse one by two equals 30 degrees

In simple words:

• Trigonometric function gives ratio from angle

• Inverse trigonometric functions give angle from ratio

If
y equals sin x
then
x equals sin inverse y

Similarly, we have inverse of cosine, tangent, cotangent, secant, and cosecant.

These are also called:

• arc sine

• arc cosine

• arc tangent

List of Inverse Trigonometric Functions

There are six main Inverse trigonometric functions:

• sin inverse x

• cos inverse x

• tan inverse x

• cot inverse x

• sec inverse x

• cosec inverse x

These functions are commonly written as:

• sin⁻¹ x

• cos⁻¹ x

• tan⁻¹ x

• cot⁻¹ x

• sec⁻¹ x

• cosec⁻¹ x

Important point: The power minus one does not mean reciprocal. It represents inverse.

Domain and Range of Inverse Trigonometric Functions

One of the most important topics in Inverse trigonometric functions Class 12 is understanding domain and range.

Since trigonometric functions are not one to one in their full domain, we restrict them to make them invertible.

1. Sine Inverse Function

• Domain: minus one to one

• Range: minus pi by two to pi by two

2. Cosine Inverse Function

• Domain: minus one to one

• Range: zero to pi

3. Tangent Inverse Function

• Domain: all real numbers

• Range: minus pi by two to pi by two

4. Cotangent Inverse Function

• Domain: all real numbers

• Range: zero to pi

5. Secant Inverse Function

• Domain: x less than or equal to minus one or x greater than or equal to one

• Range: zero to pi except pi by two

6. Cosecant Inverse Function

• Domain: x less than or equal to minus one or x greater than or equal to one

• Range: minus pi by two to pi by two except zero

These restrictions are extremely important for solving equations correctly.

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Inverse Trigonometric Functions Table

Students often look for a clear inverse trigonometric functions table for quick revision. Below is a helpful summary:

This inverse trigonometric functions table helps in solving direct questions quickly in exams.

You should memorize:

• sin inverse one by two equals pi by six

• cos inverse one by two equals pi by three

• tan inverse one equals pi by four

Principal Value Branch

Another key concept in Inverse trigonometric functions is principal value.

Since trigonometric functions repeat their values, inverse functions select only one value. That value is called principal value.

For example:

sin 30 degrees equals one by two
sin 150 degrees also equals one by two

However:

sin inverse one by two equals pi by six

It does not give 150 degrees because of the restricted range.

Properties of Inverse Trigonometric Functions

Understanding properties makes solving equations easier.

Basic Identities

• sin inverse x plus cos inverse x equals pi by two

• tan inverse x plus cot inverse x equals pi by two

• sec inverse x plus cosec inverse x equals pi by two

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Negative Argument Property

• sin inverse minus x equals minus sin inverse x

• tan inverse minus x equals minus tan inverse x

However:

• cos inverse minus x equals pi minus cos inverse x

These identities are frequently asked in Class 12 board exams.

Graphs of Inverse Trigonometric Functions

Graphs help in visual understanding.

For example:

• Graph of sin inverse x lies between minus pi by two and pi by two

• Graph of cos inverse x lies between zero and pi

• Graph of tan inverse x approaches asymptotes at minus pi by two and pi by two

Graph related questions are common in CBSE exams.

Solved Examples of Inverse Trigonometric Functions

Let us solve some examples to make the concept clear.

Example 1

Find sin inverse root three by two.

Solution:

We know from the inverse trigonometric functions table:

sin inverse root three by two equals pi by three

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Example 2

Find cos inverse minus one by two.

Solution:

cos inverse minus x equals pi minus cos inverse x

So,

cos inverse minus one by two equals pi minus pi by three
equals two pi by three

Example 3

Prove that
sin inverse x plus cos inverse x equals pi by two

Let sin inverse x equals theta

Then sin theta equals x

We know cos theta equals root one minus x square

Hence cos inverse x equals pi by two minus theta

Therefore sum equals pi by two

Hence proved.

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Application of Inverse Trigonometric Functions

Inverse trigonometric functions are not just exam topics. They are used in:

• Calculus

• Integration

• Differentiation

• Engineering mathematics

• Physics problems

• Navigation

• Architecture

In integration, many results directly involve inverse functions such as:

• Integration of one by root one minus x square equals sin inverse x

• Integration of one by one plus x square equals tan inverse x

This makes them extremely important in higher mathematics.

Important Formulas of Inverse Trigonometric Functions

Here are some key formulas students must remember:

• sin inverse x plus sin inverse y equals sin inverse x root one minus y square plus y root one minus x square

• tan inverse x plus tan inverse y equals tan inverse x plus y divided by one minus xy

Also,

• tan inverse x plus tan inverse one by x equals pi by two when x is positive

These formulas are frequently used in proof based questions.

Difference Between Trigonometric and Inverse Trigonometric Functions

Let us quickly understand the difference.

Trigonometric functions:

• Input is angle

• Output is ratio

Inverse trigonometric functions:

• Input is ratio

• Output is angle

This conceptual clarity helps students avoid confusion.

Common Mistakes Students Make

Students often make these errors:

• Confusing sin inverse x with one by sin x

• Ignoring domain restrictions

• Forgetting principal value

• Using degrees instead of radians

Remember, in Class 12 mathematics, answers are mostly expressed in radians.

Strategy to Score Full Marks in Inverse Trigonometric Functions

Here are some smart tips:

• Memorize the inverse trigonometric functions table properly

• Practice identity proofs daily

• Understand range restrictions clearly

• Solve previous year questions

• Revise formulas regularly

Consistency is the key to mastering Inverse trigonometric functions.

Why PlanetSpark Helps in Learning Inverse Trigonometric Functions

Mathematics becomes easier when concepts are explained clearly. At PlanetSpark, students:

• Learn step by step explanations

• Practice interactive questions

• Build strong conceptual foundation

• Get doubt solving support

• Improve logical thinking skills

With proper guidance, even complex topics like Inverse trigonometric functions become simple.

Quick Revision Notes

Before exams, revise these points:

• sin inverse x range is minus pi by two to pi by two

• cos inverse x range is zero to pi

• tan inverse x range is minus pi by two to pi by two

• sin inverse x plus cos inverse x equals pi by two

• Always check domain before solving

These small steps can improve your accuracy significantly.

Conclusion

To summarize, Inverse trigonometric functions are an essential part of Class 12 mathematics. They help us find angles from given trigonometric ratios and play a major role in calculus and higher mathematics.

Understanding what is inverse trig functions, memorizing the inverse trigonometric functions table, learning domain and range, and practicing identities will ensure strong performance in exams.

With regular practice and conceptual clarity, this chapter becomes scoring and interesting. So, start practicing today and build confidence in Inverse trigonometric functions Class 12.