Sin 1

There are three major ratios in trigonometry upon which the trigonometric functions and formulas are based. The sine function is one of them. The sine function (sin) of an angle gives the ratio of the perpendicular (opposite side of the angle) to the hypotenuse. Similarly, the inverse sine function (sin-1) gives the ratio of the hypotenuse to the perpendicular of an angle. Sin 1 in radians has a value of 0.8414709848. The basic angles which are required very frequently are 0,30,45,60,90 degrees. They can also be expressed in radians as π/2, π/3, π/4, π/6, π, etc.

Sin 1 in Radians

In radians, the value of sine 1 is 0.8414709848.

We know, π/3 = 1.047198≈1

\[Sin (\frac{\pi}{3}) = \frac{\sqrt{3}}{2}\] 

Sin π = 0

Now using these data, we can write;

\[sin 1 = sin \frac{\pi}{3} − (\frac{\pi}{3} − 1)\]

⟹ sin1 = \[sin \frac{\pi}{3} cos(\frac{\pi}{3} − 1)−cos \frac{\pi}{3} sin(\frac{\pi}{3} − 1)\]

The angle π/3−1=0.047198 is a very small angle.

We know that, for small angles θ,

Sinθ ≈ θ and cos⁡θ ≈ 1

hence,

Sin 1 ≈ \[(\frac{\sqrt{3}}{2} \times 1) − \frac{1}{2} \times (\frac{\pi}{3} − 1)\]

Therefore, sin1 ≈ 0.842427

How to find the Value of Sine 1?

The sine of an angle can take its argument as either radian or degrees. The rule is radian measurement.

We know, π radian = 180 degree

therefore, 1 rad = 180/π degree

1 rad = 57.2957795131 degree

In degrees, we know that,

sin 0° = 0, sin 90° = 1

In radians,

sin 0 = 0 and sin (π/2)=1

Now, π = 3.14159265359, π/2=1.5707963268

Therefore, 

sin (1.5707963268)= 1, when the angle is taken as radian

So,

sin (1) = 0.8414709848, when the angle is taken as radian

sin (57.2957795131) = 0.8414709848, when the angle is taken as degree

Value of Sin 1 from Taylor's Series

According to Taylor's Series, we know that

\[f(x) = f(a) + \frac{f’(a)}{1!} (x-a) + \frac{f’’(a)}{2!} (x-a)^2 + \frac{f’’’(a)}{3!} (x-a)^3 + ………. \]

From this series, we can find out the value of Sin 1. 

Hence, putting f(x) = sin 1 we get-

\[sin 1 = 1 - \frac{1}{3!} + \frac{1}{5!} + \frac{1}{7!} + ………. \]

Or \[sin 1 = 1 - \frac{1}{6} + \frac{1}{120} + \frac{1}{5040} + ………. \]

Or, Sin 1 ≈ 0.82

Thus, we can find out the value of Sin 1 from Taylor's Series. 

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Sine of 1 in Terms of Π

The angle, whose sine is 1, is the inverse function of sin 1. As sine of the angle 90° is 1, it is equal to the function sin 1. So, the inverse function of sin 1 is denoted as 90° or π/2. It is the highest value of the sine function. 

Before knowing about the value of the inverse of sin 1 [sin-1 (1)], let us discuss the meaning of the inverse of sin.

Inverse Sine Function

Arcsine which is also known as the inverse sine function is the inverse of the sine function. The representation of the inverse of sine is (sine-1). The inverse sine function is used to determine the value of the angle.

θ = sin-1 (opposite side of angle θ/ hypotenuse)

The principal branch value of angle theta is. This means that the value of theta will lie between 

To understand the concept of sin inverse, let us consider an example.

Example 1 - In a triangle, the measure of the three sides hypotenuse = 5 cm, perpendicular side = 4 cm and base = 3 cm. With help of the sides given, find the value of angle θ.

Ans. Given: hypotenuse = 5 cm, perpendicular side = 4 cm and base = 3 cm

We know that,

sin θ = \[\frac{\text{opposite side of angle} \theta }{hypotenuse}\]

θ = sin-1 

θ = sin-1 (4/ 5)

θ = sin-1 (0.8)

θ = 0.92

Therefore, sin θ = 0.8

sin (0.92) = 0.8

Value of the Inverse of Sin 1 (Sin -1 1)

The inverse sin of 1, i.e., sin-1 (1) gives a very unique value for the inverse of the sine function. Sin-1 (x) will give us the angle whose sine is x, which means the ratio of the perpendicular to the hypotenuse is x. Hence, sin-11 (1) is equal to the angle whose value of the sine function is 1.

We know,

Sin 90 = 1

Therefore,

sin-11(1) = 90 ( when angle is in degrees)

sin-1(1) = π/2 (when angle is in radian)

Since the inverse of sin-1 (1) is 90° or π/2, the maximum value of the sine function is denoted by ‘1’. Therefore, for every 90 degrees the same will happen, such as at π/2, 3π/2, and so on.

So by this, we can say that,

sin-1(1) = π/2 + 2πn (n denotes any integer)

Solved Example

1. Find out the Value of 4sin-11.

Solution: 

Suppose, x = sin-11 

Then, sinx = 1

We know, sin π/2 = 1

So, here x = π/2

Now, 4 sin-11 = 4 * π/2    = 2π 

2. Calculate the Value of 2 sin 1 in Radians.

Solution:

As we know that the value of sin 1 in radians is equal to 0.84.

Therefore, 2 sin 1 = 2 * 0.84   = 1.68 

Did You know?

Sine function denotes the ratio of the largest side and one adjacent side of the angle 90°. The inverse function of sine (sin-1) is used to find out the angle opposite to these two sides of the right-angled triangle.