Inverse trigonometric functions are used to find an angle when the value of a trigonometric ratio is known. The most common inverse functions are inverse sine, inverse cosine, and inverse tangent, written as sin⁻¹x, cos⁻¹x, and tan⁻¹x. These functions are especially important when solving trigonometric equations and finding unknown angles.
This guide explains important inverse trigonometric functions formulas, identities, domains, ranges, examples, and common mistakes in a student-friendly way.
What Are Inverse Trigonometric Functions?
An inverse trigonometric function reverses the operation of a trigonometric function. For example:
y = sin⁻¹x means sin y = x.
Therefore, if:
sin 30° = 1/2
then:
sin⁻¹(1/2) = 30°
Similarly:
cos⁻¹(√3/2) = 30°
and
tan⁻¹(1) = 45°
Inverse functions return an angle from a given trigonometric ratio. Their principal-value ranges are restricted so that the functions are one-to-one.
Important Inverse Trigonometric Functions Formulas
The three most frequently used inverse trigonometric functions are:
| Function | Meaning |
| sin⁻¹x | Inverse sine of x |
| cos⁻¹x | Inverse cosine of x |
| tan⁻¹x | Inverse tangent of x |
Some important inverse trigonometric identities are:
sin⁻¹x + cos⁻¹x = π/2
tan⁻¹x + cot⁻¹x = π/2
For appropriate values of x and y:
tan⁻¹x + tan⁻¹y = tan⁻¹[(x + y)/(1 − xy)]
and
tan⁻¹x − tan⁻¹y = tan⁻¹[(x − y)/(1 + xy)]
The addition and subtraction formulas require attention to the correct principal-value range; they should not be applied blindly without checking the resulting angle.
Domain and Range of Inverse Trigonometric Functions
Knowing the domain and range is essential because inverse trigonometric functions use specific principal-value branches.
| Function | Domain | Principal-value Range |
| sin⁻¹x | −1 ≤ x ≤ 1 | −π/2 ≤ y ≤ π/2 |
| cos⁻¹x | −1 ≤ x ≤ 1 | 0 ≤ y ≤ π |
| tan⁻¹x | All real numbers | −π/2 < y < π/2 |
| cot⁻¹x | All real numbers | 0 < y < π |
| sec⁻¹x | x ≤ −1 or x ≥ 1 | [0, π], y ≠ π/2 |
| cosec⁻¹x | x ≤ −1 or x ≥ 1 | [−π/2, π/2], y ≠ 0 |
The standard principal-value domains and ranges are important when simplifying expressions involving inverse functions.
Basic Inverse Trigonometric Identities
Several identities are useful for simplifying expressions.
Sine and Cosine
sin(sin⁻¹x) = x, for −1 ≤ x ≤ 1
cos(cos⁻¹x) = x, for −1 ≤ x ≤ 1
However:
sin⁻¹(sin x) = x
is only directly true when:
−π/2 ≤ x ≤ π/2
Similarly:
cos⁻¹(cos x) = x
is only directly true for:
0 ≤ x ≤ π
This distinction occurs because inverse functions return values only within their principal-value ranges.
Tangent
tan(tan⁻¹x) = x
for every real number x.
Also:
tan⁻¹(tan x) = x
when:
−π/2 < x < π/2
These restrictions are important when solving inverse trigonometric problems.
Important Reciprocal Identities
The following relationships can also be useful:
sin⁻¹x = cosec⁻¹(1/x)
cos⁻¹x = sec⁻¹(1/x)
tan⁻¹x = cot⁻¹(1/x)
where the expressions are defined and the chosen conventions for inverse functions are consistent.
Students should be careful with reciprocal identities because principal-value conventions matter.
Inverse Trigonometric Functions Examples
Example 1: Inverse Sine
Evaluate:
sin⁻¹(1/2)
We know:
sin(π/6) = 1/2
Therefore:
sin⁻¹(1/2) = π/6
or 30°.
Example 2: Inverse Cosine
Evaluate:
cos⁻¹(1/2)
Since:
cos(π/3) = 1/2
we get:
cos⁻¹(1/2) = π/3
or 60°.
Example 3: Inverse Tangent
Evaluate:
tan⁻¹(1)
Since:
tan(π/4) = 1
therefore:
tan⁻¹(1) = π/4
or 45°.
A More Important Example: Principal Values
Consider:
sin⁻¹(sin 3π/4)
It may seem that the answer is 3π/4, but this is incorrect because the principal range of sin⁻¹x is:
[−π/2, π/2]
Now:
sin(3π/4) = sin(π/4)
and π/4 lies inside the principal range.
Therefore:
sin⁻¹(sin 3π/4) = π/4
This is one of the most common concepts students need to understand when working with inverse trigonometric functions.
Inverse Trigonometric Functions vs Reciprocal Functions
A common mistake is confusing an inverse function with a reciprocal.
For example:
sin⁻¹x means inverse sine of x.
It does not mean:
1/sin x
The reciprocal of sin x is:
cosec x = 1/sin x
Likewise:
cos⁻¹x is inverse cosine, while 1/cos x = sec x.
This distinction is fundamental in inverse trigonometry.
Common Mistakes to Avoid
1. Confusing inverse and reciprocal functions
Do not treat sin⁻¹x as 1/sin x.
2. Ignoring principal-value ranges
An inverse trigonometric function does not always return the original angle. Always check its principal range.
3. Forgetting the domain
For sin⁻¹x and cos⁻¹x, the input must satisfy:
−1 ≤ x ≤ 1
4. Applying tangent addition formulas without checking the angle
Inverse tangent addition and subtraction formulas can require adjustments depending on the quadrant and principal-value range.
5. Mixing degrees and radians
Always check whether the question expects the answer in degrees or radians.
How to Learn Inverse Trigonometric Formulas
A useful approach is to learn the formulas in groups rather than memorizing a long list at once.
First, memorize:
- sin⁻¹x + cos⁻¹x = π/2
- The domains and ranges of sin⁻¹x, cos⁻¹x, and tan⁻¹x
- The basic composite identities
- Standard values such as sin⁻¹(1/2), cos⁻¹(1/2), and tan⁻¹(1)
Then practice simplifying expressions and evaluating inverse functions without relying immediately on a calculator.
Key Takeaways
- Inverse trigonometric functions find angles from known trigonometric ratios.
- The main functions are sin⁻¹x, cos⁻¹x, and tan⁻¹x.
- Domain and principal-value range restrictions are essential.
- sin⁻¹x does not mean 1/sin x.
- Always check the principal range when simplifying expressions such as sin⁻¹(sin x).
- Practice standard values and identities to improve speed and accuracy.
Conclusion
Inverse trigonometric functions formulas are essential for solving problems involving unknown angles and trigonometric equations. Understanding the formulas alone is not enough; students should also learn the domain, range, principal values, identities, and correct application of each function.
The most important habit is to check the principal-value range before accepting an answer. With regular practice, inverse trigonometric identities become much easier to recognize and apply.
For additional mathematics learning, students can explore Digital Libraries for formulas, worksheets, practice questions, and other educational resources.
FAQ
What are inverse trigonometric functions?
Inverse trigonometric functions find an angle when the value of a trigonometric ratio is given.
What is the inverse sine formula?
The inverse sine function is written as sin⁻¹x and returns the principal angle whose sine is x.
What is the inverse cosine formula?
cos⁻¹x gives the principal angle whose cosine is x, with x restricted to −1 ≤ x ≤ 1.
What is the inverse tangent formula?
tan⁻¹x gives the principal angle whose tangent is x. Its domain is all real numbers.
What is the difference between sin⁻¹x and 1/sin x?
sin⁻¹x represents inverse sine, while 1/sin x represents the reciprocal of sine, called cosecant.