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Inverse Sine function or the arcsine function is the inverse of the sine function. We know that the sine of an angle or sine function is equal to the ratio of the opposite side of a triangle and hypotenuse. Therefore, the sine inverse of the same ratio will give the measure of the angle (θ). It is used to measure the angle, which is done with the basic trigonometric ratios from the right angle. In major cases, the inverse of the sine function is represented as sin-1. The ‘-1’ representation does not convey that the sine is elevated to the power of (-1); it is just an indication used.
Read Also: Trigonometric Functions
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Key Terms: Inverse Sine, Sine Function, Trigonometric Functions, Trigonometric Ratios, Inverse Trigonometric Functions, Derivative, Arcsine Function, Ratio, Hypotenuse, Trigonometric Identities
What is Sine Function?
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The sine function in the right-angled triangle can be defined as the value equal to the opposite side to θ divided by the hypotenuse.
\(\text{Sin }\theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}}\)
What is Inverse Sine Function?
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We have already discussed the sine function in which we can determine the sin of any angle if the opposite side and hypotenuse are known to us. The inverse of the sine function or sine-1 can find the resultant angle when the opposite angle of θ is divided by the hypotenuse. The angle is produced when the ratio when the opposite angle θ is divided by the hypotenuse. It is also denoted or written as arcsin.
Thus, Sin Inverse is denoted by sin-1 or arcsin.
\(\theta = Sin^{-1} = \frac{\text{Opposite Side}}{\text{Hypotenuse}}\)
Inverse Sine Function
Check Important Notes for Inverse Trigonometric Functions
Inverse Sine Formula
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For figuring out the formula of the sine function, we need to make an assumption. The assumption is made of the depth of the seabed, which is denoted by x, and its depth has to be measured from the bottom of the sea. For the same, the parameters given are
- The length of the cable
- The measure of the angle made by the cable with the seabed
For measuring both the above-listed parameters, we will be using the sine function,
The given angle of the cable is given as 39°, and the length is given as 40 m,
Therefore, Sine 39° = Opposite Side/ Hypotenuse
Sine 39° = x/ 40
x = 40 × Sine 39°
x = 40 × 0.6293
x = 25.172 cm
Thus, the depth of the seabed is 25.172 cm.
Thus, we can summarize the formulas for the sine function and inverse sine function.
- The trigonometric sine function is given by the formula:
\(\text{Sin }\theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}}\)
- The inverse sine function or the arcsin can be given by the formula:
\(Sin^{-1} = \frac{\text{Opposite Side}}{\text{Hypotenuse}} = \theta\)
Read More: Properties of Inverse Trigonometric Functions
Inverse Sine Graph
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The graph of the inverse of the sine is shown by the graph, which denoted the different values the function holds are various degrees:
Inverse Sine Graph
Check Also: Graphs of Inverse Trigonometric Functions
Inverse Sine Derivative
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The derivative of the Inverse of the Sine function is denoted in the given form:
\(\frac{d}{dx} Sine^{-1}x= \frac{1}{\sqrt{1-x^{2}}}\)
Proof: The proof of the above-given equation can be given as follows:
f(x) = sin(x) and g(x) = sin-1x
If we differentiate g(x) concerning x, we get the equation as;
g'(x) = 1/f'(g(x)) = 1/cos(sin-1x) …………….(1)
Now we know the inverse of the sine function, y = Sin-1x, which can also be written as;
sin y = x …………(2)
Hence, we can summarise the derivative as:
g'(x) = 1/cos y (from eq.1)
With our previous knowledge about trigonometric identities, we can write the equation as,
cos2y + sin2y = 1
The above-given equation can also be noted down as;
cos y = √(1-sin2y) = √(1 – x2) (since sin y = x)
Now, putting this value again in the derivative
g'(x) = 1/cos y = 1/√(1 – x2)
Thus,
d/dx(sin-1x) = 1/√(1 – x2)
Hence proved.
Read More: Derivative of Inverse Trigonometric Functions
Inverse Sine Value Table
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The value of the inverse sine function is given with various angles concerning the sine function and the inverse of the sine function.
| θ | Sin-1 or Arcsin(θ) (in Radian) | Sin-1 or Arcsin(θ) (in Degree) |
|---|---|---|
| -1 | -π/2 | -90° |
| -√3/2 | -π/3 | -60° |
| -√2/2 | -π/4 | -45° |
| -1/2 | -π/6 | -30° |
| 0 | 0 | 0° |
| 1/2 | π/6 | 30° |
| √2/2 | π/4 | 45° |
| √3/2 | π/3 | 60° |
| 1 | π/2 | 90° |
Read More: Trigonometry Table
Things to Remember
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- The inverse sine function, also called as the arcsine function, is the inverse of the sine function. The inverse of the sine function is represented as sin-1.
- Sin-1 is the inverse of the sine function whereas 1/sin x shows the reciprocal of the sine function, which is moreover equal to the cosecant function.
- The arcsine is employed to analyse the angle whose sine value is equal to the ratio of its opposite side and hypotenuse. Therefore, if we know the extent of the opposite side and hypotenuse, we can find the extent of the angle.
- The formula for the trigonometric function sine is Opposite Side divided by Hypotenuse.
- The formula for inverse sine function is given as: sin-1 (Opposite side/ hypotenuse) = θ.
- The derivative of the Inverse of the Sine function is denoted in the given form: d/dx Sine-1 x= 1/√(1-x2)
Sample Questions
Ques. Find the values of sin (cos−13/5) (3 Marks)
Ans. Let, cos−1 3/5 = θ
Thus, cos θ= 3/5
With that, we can say that sin θ= √1- cos2θ
= √1- 9/25
= √16/25
= 4/5
Thus the sin (cos−13/5) values to 4/5
Ques. In a triangle, ABC, AB= 4.9m, BC=4.7 m, CA=2.5 m and angle B = 30°. Find Sin−1. (3 Marks)
Ans. Sin 30° = Opposite / Hypotenuse
Sin 30° = 2.5 / 4.7
Sin 30° = 0.53
So, Sin-1 (Opposite / Hypotenuse) = 30°
Sin-1 (0.53) = 30°
Ques. In a triangle, ABC, AB= 90m, BC=45 m, CA=28 m and angle B = 29°. Find Sin−1 (3 Marks)
Ans. Sin 29° = Opposite / Hypotenuse
Sin 29° = 28/ 45
Sin 29° = 0.62
So, Sin-1 (Opposite / Hypotenuse) = 30°
Sin-1 (0.62) = 29°
Ques. What will be the principal value of sin-1 (-1/2 ) in radians and degrees? (3 Marks)
Ans. Consider y= sin-1 ( - 1/2 ), thus, sin y = - 1/2
We know that the range of the principal value of sin-1x is [- π/2 , π/2 ], so, we must find y ∈ [- π/2 , π/2 ] such that sin y = - 1/2 .
It is clear that y = - π /6.
Therefore, the principal value of sin-1 ( - 1/2 ) is – π/6 which corresponds to −30°.
Ques. Simplify cos(arcsin(x)) (3 Marks)
Ans. Let, z = cos ( arcsin x ) and y = arcsin x
So that, z = cos y.
According to the theorem, y = arcsin x may also be written as
sin y = x with - π / 2 ≤ y ≤ π / 2
Also, sin2y + cos2y = 1
Now, we will substitute sin y by x and solve for cos y to obtain
cos y = + or - √ (1 - x2)
However, - π / 2 ≤ y ≤ π / 2 so that cos y is positive
z = cos y = cos(arcsin x) = √ (1 - x2)
Ques. What is arcsin? (3 Marks)
Ans. The arcsine is the inverse of the sine function. It is employed to analyse the angle whose sine value is equal to the ratio of its opposite side and hypotenuse. Therefore, if we know the extent of the opposite side and hypotenuse, we can find the extent of the angle.
Ques. What is the distinction between sin−1 and 1/sin x? (2 Marks)
Ans. Sin-1 is the inverse of the sine function. -1 here does not exemplify the proponent. Arcsin α refers to the arc whose sine is α. Whereas 1/sin x shows the reciprocal of the sine function, which is also proportional to the cosecant function.
Ques. Is the sine inverse equal to the cosec function? (3 Marks)
Ans. Sine inverse of arcsin is the inverse of the sine function, which returns the value of angle for which sine function is equal to opposite side and hypotenuse ratio. It generates the value of the angle. But the sec function is the mutual of the one function and is not equal to the one inverse.
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