In mathematics, the inverse trigonometric functions are the inverse functions of trigonometric functions. Specifically, they are inverse of the sine, cosine, tangent, cotangent, secant and cosecant functions, and are used to obtain an angle from any of the angle’s trigonometric ratios.
What are Trigonometric Functions?
A function f : A → B is said to be invertible if f is bijective (i.e., one-one and onto). The inverse of the function f is denoted by f : B → A such that f-1(y) = x if f(x) = y, ∀ x ∈A, y ∈B. As trigonometric functions are many-one, their inverse doesn’t exist. But they become one-one and onto by restricting their domains. Therefore, all the restrictions are required so that the inverse of the concerned trigonometric functions do exist. If these restrictions are removed, the terms will represent inverse trigonometric relations and not the functions. Note that the inverse trigonometric functions are also called inverse circular functions.
Domains and Ranges of Inverse Trigonometric Functions:
| Inverse trigonometric function | Domain | Principal value range |
|---|---|---|
| sin-1 (x) | [-1, 1] | [-\(\frac{\pi}{2} \),\(\frac{\pi}{2} \)] |
| cos-1 (x) | [-1, 1] | [0,\(\pi\)] |
| tan-1 (x) | R | (-\(\frac{\pi}{2} \),\(\frac{\pi}{2} \)) |
| cosec-1 (x) | R-(-1,1) | [-\(\frac{\pi}{2} \),\(\frac{\pi}{2} \)]-{ 0} |
| sec-1 (x) | R-(-1,1) | [0,\(\pi \)]-{\(\frac{\pi}{2} \)} |
| cot-1 (x) | R | (0,\(\pi \)) |
How to Find Principal Value?
The numerically smallest angle is known as the principal value.
For finding the principal value, following algorithm can be followed:
Step 1: First, draw a trigonometric circle and mark the quadrant in which the angle may be lie.
Step 2: Select anti-clockwise direction for 1st and 2nd quadrant in which the angle may be lie.
Step 3: Find the angles in the first rotation.
Step 4: Select the numerically least (magnitude-wise) angle among these two values. The angle thus found will be the principal value.
Step 5: In case, two angles one with a positive sign and the other with the negative sign qualify for the numerically least angle then, it is the convention to select the angle with positive sign as principal value.
Elementary Properties of Inverse Trigonometric Functions:
1. Identities
- sin⁻¹(x) + cos⁻¹(x) = π/2
- tan⁻¹(x) + cot⁻¹(x) = π/2
- sec⁻¹(x) + cosec⁻¹(1/x) = π/2 (for |x| ≥ 1)
2. Inverse Relations
- sin(sin⁻¹(x)) = x for x ∈ [−1, 1]
- cos(cos⁻¹(x)) = x for x ∈ [−1, 1]
- tan(tan⁻¹(x)) = x for x ∈ ℝ
3. Odd-Even Properties
- sin⁻¹(−x) = −sin⁻¹(x)
- tan⁻¹(−x) = −tan⁻¹(x)
- cos⁻¹(−x) = π − cos⁻¹(x)
Some Important Questions on Inverse Trigonometric Functions:
Q1. Find the value of sin-1(1/2) + cos-1(1/2).
Solution: We use the identity tan⁻¹x + cot⁻¹x = π/2, for all x ∈ ℝ
So, tan⁻¹(1) + cot⁻¹(1) = π/2
Answer = π/2
Q5. Find the value of cos-1(–1).
Solution: Let y=cos⁻¹(-1)
cosy=-1
\(cosy=cosπ \) \(\Rightarrow y=\pi \) We know that the principal value range of cos⁻¹x is [0, π] \[ y \in [0, π] \]
Hence,Answer = π
🔍 Frequently Asked Questions (FAQs) on Inverse Trigonometric Functions
Question 1: What are inverse trigonometric functions?
Answer: Inverse trigonometric functions are the inverse functions of the standard trigonometric functions (sine, cosine, tangent, etc.). They help us find the angle when the value of the trigonometric ratio is known.
Question 2: What are the main inverse trigonometric functions?
Answer: There are six inverse trigonometric functions: cot⁻¹(x) , sin⁻¹(x) , cos⁻¹(x), tan⁻¹(x), cosec⁻¹(x), sec⁻¹(x).
Question 3: What is the domain and range of sin⁻¹(x)?
Answer: Domain: [-1, 1]
Range:\( [- \frac{\pi}{2}, \frac{\pi}{2}] \)
