Inverse trigonometric functions are the inverse functions of the basic trigonometric functions: sine, cosine, and tangent. To define these inverse functions, we need to restrict the domain and range of the original functions so that they become one-to-one and onto.
Here are the domain and range for each of the primary inverse trigonometric functions:
- Inverse Sine function ($sin^{-1}$ or arcsin):
Domain: $-1 \leq x \leq 1$, as the sine function takes values in this range.
Range: $-\pi/2 \leq y \leq \pi/2$, which corresponds to the interval over which sine is a strictly increasing function.
- Inverse Cosine function ($cos^{-1}$ or arccos):
Domain: $-1 \leq x \leq 1$, as the cosine function takes values in this range.
Range: $0 \leq y \leq \pi$, which corresponds to the interval over which cosine is a strictly decreasing function.
- Inverse Tangent function ($tan^{-1}$ or arctan):
Domain: $-\infty < x < \infty$, as the tangent function takes values over the entire real line.
Range: $-\pi/2 < y < \pi/2$, which corresponds to the interval over which tangent is a strictly increasing function.
There are also three other inverse trigonometric functions corresponding to the reciprocal functions of sine, cosine, and tangent: inverse cosecant, inverse secant, and inverse cotangent. Their domain and range are as follows:
- Inverse Cosecant function ($csc^{-1}$ or arccsc):
Domain: $x \geq -1$ or $x \geq 1$, as the cosecant function takes values in this range.
Range: $-\pi /2 \leq y < 0$ or $0 < y \leq \pi/2$, which corresponds to the interval over which cosecant is a strictly decreasing function.
- Inverse Secant function ($sec^{-1}$ or arcsec):
Domain: $x \leq -1$ or $x \geq 1$, as the secant function takes values in this range.
Range: $0 \leq y < \pi/2$ or $\pi/2 < y \leq \pi$, which corresponds to the interval over which secant is a strictly decreasing function.
- Inverse Cotangent function ($cot^{-1}$ or arccot):
Domain: $-\infty < x < \infty$, as the cotangent function takes values over the entire real line.
Range: $0 < y < \pi$, which corresponds to the interval over which cotangent is a strictly decreasing function.
How to remember it
- The domain of the inverse function is the range of the original trigonometric function
- The range of the Inverse function is the smallest interval in which it becomes one-one and onto.lets see how we can remember it
Lets first rewrite the table in below manner
We can use below tricks to remember the range base on this
- We can clearly see that inverse of sin, cosec and tan have a similar type of range while inverse of cos ,sec and cot having similar type of range
- sin and cosec but it exclude 0 as then it will become undefined, Similar for cos and sec
- inverse of tan and cot are in different interval
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