Sin of Sin Inverse
Before going to learn what is "sin of sin inverse of x" (which is written as sin(sin^{1}x)), let us recall a few facts about the domain and range of sin and sin^{1} (which is sin inverse). We know that sine function is a function from R → [1, 1]. But sine function is NOT a bijection (as it is NOT oneone) on the domain R. Hence it cannot have inverse if its domain is R. Thus, for the sine function to be oneone, its domain is restricted to [π/2, π/2]. We also know that the domain and range of a function will be the range and domain of its inverse function respectively. Hence, the domain of the inverse of sin, which is sin^{1}, is [1, 1] and its range is [π/2, π/2]. Keeping this in mind let us see what is "sin of sin inverse of x" and "sin inverse of sin of x" here.
1.  What Is Sin of Sin Inverse of x? 
2.  How To Calculate Sin of Sin Inverse of x? 
3.  How To Calculate Sin Inverse of Sin of x? 
4.  FAQs on Sin of Sin Inverse of x 
What Is Sin of Sin Inverse of x?
Let us examine a few examples given below to understand the concept of "sin of sin inverse of x". We can calculate each of the following using the calculator.
 sin(sin^{1}0) = 0
 sin(sin^{1}1) = 1
 sin(sin^{1}(1/2)) = 1/2
 sin(sin^{1}(0.3)) = 0.3
 sin(sin^{1}1.2) = ERROR
Oops! Why did we get an error at the end? This is because the domain of sin^{1} (x) function is [1, 1]. It means sin of sin inverse takes only the values that lie between 1 and 1 (both inclusive). i.e., whatever number that goes inside the brackets of sin^{1}( ) should lie between 1 and 1 (both inclusive). For all the other values, the calculator throws an error. We can therefore summarize the sin of sin inverse formula as,
sin of sin Inverse Formula:
 sin (sin^{1}x) = x, whenever x ∈ [1, 1] and
 sin (sin^{1}x) is NOT defined whenever x ∉ [1, 1].
How To Calculate Sin of Sin Inverse of x?
We calculate sin of sin inverse of x using its definition mentioned in the previous section. So to calculate sin(sin^{1}x),
 See whether x lies in the interval [1, 1].
 If so, sin(sin^{1}x) = x
 Otherwise, sin(sin^{1}x) = NOT defined.
Here are few more examples on sin of sin inverse.
 sin(sin^{1}0.5) = 0.5 (as 0.5 ∈ [1, 1])
 sin(sin^{1}(0.3)) = 0.3 (as 0.3 ∈ [1, 1])
 sin(sin^{1}√2) = NOT Defined (as √2 ∉ [1, 1])
 sin(sin^{1}(1.3)) = NOT Defined (as 1.3 ∉ [1, 1])
How To Calculate Sin Inverse of Sin of x?
Before going to learn what is "sin inverse of sin of x", let us examine few examples.
 Example 1: We know that sin π/2 = 1, and so π/2 = sin^{1}(1), you can check the value of sin^{1}(1) using your calculator and you will get π/2. i.e., we have got sin^{1}(sin π/2) = π/2.
 Example 2: We know that sin π = 0, then is π = sin^{1}(0)? Can you check the value of sin^{1}(0) using your calculator? Did you get π back? No, right? You must have got sin^{1} (0) = 0. i.e., we have got sin^{1} (sin π) = 0 but sin^{1}(sin π) ≠ π. Why is this happening?
From the above two examples, we can conclude that sin^{1}(sin x) is NOT always x. In fact, sin^{1}(sin x) = x only when x lies in the interval [π/2, π/2]. Then how to find the value of sin^{1}(sin x) when x lies beyond this interval? Let us understand by an example.
Example: Find the value of sin^{1}(sin 8) (Note that 8 is in radians here).
Solution:
Here the angle is 8 radians.
Step  1: Find two consecutive multiples of π between which the given angle lies.
We have 2π < 8 < 3π (as π = 3.142)
Note that among these multiples of π, one(2π) is an even multiple of π and the other(3π) is an odd multiple of π.
Step  2: Find "odd multiple of π  x" and "x  even multiple of π".
odd multiple of π  x = 3π  8 ≈ 1.42
x  even multiple of π = 8  2π ≈ 1.72
Step  3: See which among the above lies between [π/2, π/2].
We have [π/2, π/2] = [1.57, 1.57], and among the above two values (1.42 and 1.72), 1.42 lies in this interval.
Thus, sin^{1}(sin 8) = 1.42 radians.
Important Notes:
Here are some important notes regarding sin of sin inverse and sin inverse of sin.
 sin (sin^{1}x) = x, ONLY when x ∈ [1, 1]. This is because the domain of sin^{1} is [1, 1].
 sin^{1}(sin x) = x, ONLY when x ∈ [π/2, π/2]. This is because the domain of sin to have inverse is [π/2, π/2].
Related Topics:
You might find the following topics helpful while reading this article "sin of sin inverse".
Examples on Sin of Sin Inverse of x

Example 1: Find the exact values of a) sin(sin^{1} (2.3)) b) sin(sin^{1} (0.3)).
Solution:
We know that sin of sin inverse of x [ sin(sin^{1} (x)) ] = x only when x lies in the interval [1, 1]. Otherwise sin(sin^{1} (x)) is NOT defined.
Using this,
a) sin(sin^{1} (2.3)) is NOT defined as 2.3 is NOT in [1, 1].
b) sin(sin^{1} (0.3)) = 0.3 as 0.3 is in [1, 1].
Answer: a) sin(sin^{1} (2.3)) is NOT defined and b) sin(sin^{1} (0.3)) = 0.3.

Example 2: Find the exact values of a) sin^{1}(sin (π/2)) b) sin^{1}(sin (7π/6)).
Solution:
Using sin of sin inverse formula,
a) sin^{1}(sin (π/2)) = π/2, as π/2 ∈ [π/2, π/2].
b) 7π/6 ∉ [π/2, π/2]. Thus sin inverse of sin 7π/6 is NOT 7π/6.
We find two consecutive multiples of π between which 7π/6 lies.
Two such multiples of π are π and 2π.
We find
odd multiple of π  x = π  7π/6 = π/6 ( ≈ 0.52)
x  even multiple of π = 7π/6  2π = 5π/6 (≈ 2.6)
Among these two, π/6 ( ≈ 0.52) lies in [π/2, π/2].
Thus, sin^{1}(sin (7π/6)) = π  7π/6 = π/6.
Answer: a) sin^{1}(sin (π/2)) = π/2 and b) sin^{1}(sin (7π/6)) = π/6.

Example 3: Evaluate sin(sin^{1}(√2/2))  sin(sin^{1}(√2/2)).
Solution:
We know that sin of sin inverse of x is x when x ∈ [1, 1].
Since √2/2 ≈ 0.707 ∈ [1, 1],
sin(sin^{1}(√2/2))  sin(sin^{1}(√2/2))
= √2/2  (√2/2)
= 2√2/2
= √2
Answer: sin(sin^{1}(√2/2))  sin(sin^{1}(√2/2)) = √2.
FAQs on Sin of Sin Inverse of x
What is Sin of Sin Inverse of x?
sin of sin inverse of x is a trigonometric function denoted as, sin (sin^{1}x) = x, when x ∈ [1, 1]. If x ∉ [1, 1] then sin (sin^{1}x) is undefined.
What is Sin Inverse of Sin of x and Sin of Sin Inverse of x in Trigonometry?
sin inverse of sin x can be given as, sin^{1}(sin x) = x, only when x ∈ [π/2, π/2]. If x ∉ [π/2, π/2], then we will find two consecutive multiples of π between which x lies. Then we find "odd multiple of π  x" and "x  even multiple of π". Only one among these lies in [π/2, π/2] and that is the value of sin^{1}(sin x). Also, sin of sin inverse of x can be given as, sin (sin^{1}x) = x, when x ∈ [1, 1]. If x ∉ [1, 1] then sin (sin^{1}x) is undefined.
Is Sin of Sin Inverse of x equals x always?
No, sin (sin^{1}x) is NOT always x. sin of sin inverse can be given as, sin (sin^{1}x) = x only when x is in [1, 1].
How to Find Inverse Sin of Sin of x?
sin^{1}(sin x) is NOT x always. sin^{1}(sin x) = x only when x is in [π/2, π/2].
What is Sin of Sin Inverse of 1?
sin (sin^{1}x) = x when x lies in [1, 1]. Since 1 lies in this interval, sin(sin^{1}1) = 1.
Is the Mathematical Notation of Sin Inverse of Sin of x Same as Notation of Sin of Sin Inverse of x?
sin and sin inverse are functions that are inverses of each other and have different mathematical notations. sin inverse of sin of x is written as sin^{1}(sin x), whereas sin of sin inverse x is written as sin(sin^{1}x).
What is Sin of Sin Inverse Value of x when x Is NOT in [1, 1]?
sin (sin^{1}x) = x, only if x lies in [1, 1]. If x does NOT lie in [1, 1], then sin^{1}(x) is NOT defined and hence sin (sin^{1}x) is also NOT defined.
What is Sin Inverse of Sin of x when x Is NOT in [π/2, π/2]?
sin^{1}(sin x) = x, only if x is in the interval [π/2, π/2]. If x is NOT in the interval, then find two multiples of π such that x lies between them. Then find "odd multiple of π  x" and "x  even multiple of π". Choose one of these angles which lies in [π/2, π/2] to be the value of sin^{1}(sin x).
What is Sin of Inverse Cos x?
To find sin of inverse cos x, first we have to convert cos^{1} into sin^{1}. Then sin(cos^{1}x) = sin(sin^{1}√(1x^{2})) = √(1x^{2}).
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