Dhristi JEE 2022-24

Concept of 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-1

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. 

FAQs on Sin 1

1. Are sin 1 and 1 sin the Same?

In trigonometry, there are some primary functions related to the angles of a right-angled triangle. The sine function is one of the 3 primary functions in trigonometry. It is defined as the ratio of the largest side and one smaller side of a right-angled triangle, which is adjacent to the 90° angle. Sin 1 means the ratio of the largest side and one adjacent side of the 90° angle is 1. On the other hand, 1 sin does not have any meaning because there is no value written after ‘sin’. Now, we can say that sin 1 has a particular value and 1 sin is meaningless. Therefore, sin 1 and 1 sin are not the same. 

2. What is the difference between sin-11 and 1/sin1?

Sine function is one of the most important primary functions of trigonometry. It is defined as the ratio of the largest side and one adjacent side of the angle 90° in a right-angled triangle. Sin 1 is the functional representation of the angle 90°, which is near about 0.84 in radians. The inverse of any trigonometric function is equal to an angle. We know that the value of sin-11 is 90°. On the other hand, 1/sin1 is calculated by dividing 1 by sin 1. 1/sin1 means the inverse ratio of the largest side and one adjacent side of the angle 90°. Hence, we can say that sin-11 is the value of an angle and 1/sin1 is the ratio of the largest side and one smaller side of a right-angled triangle. It is the main difference between sin-11  and 1/sin1. 

3. Is there any content available on the trigonometric ratio called "sin 1"?

Many articles or study materials on the trigonometric ratio called "sin 1" are available on the internet. But the best content on this topic is accessible to the students on the official webpage of Vedantu. The subject matter experts have discussed the concept behind sin 1, its conversion and its inverse. They have also given examples so that students can understand the theory of sin 1. By thoroughly going through this content, students will be able to solve the questions related to sin 1. They can even download this PDF file free of cost. 

4. Estimate the value of sin-1 (sin 10).

We know that principal branch value of sin-11  is (-π/ 2, π/ 2)

Also, sin-1 (sin x) = x

Therefore, the value of x will lie between (-π/ 2, π/ 2)

According to the question, sin-1 (sin 10).

From the above given equation, x = 10

But this value does not lie between (-π/ 2, π/ 2)

So, we find a value that lie between the principal range

Hence, 3π – 10 lies between (-π/ 2, π/ 2)

Now we can write, sin 10 = sin (3π – 10)

Thus, sin-1 (sin 10) = sin-1 [sin (3π – 10)] = 3π – 10 

5. What will be the value of 6 sin-1(1) + 8 sin-1 (1)?

Let us consider, x =sin-1 (1) 

From rearranging, sin x = 1

As we know that, sin π/2 = 1

Here, π/2 = 90°

Hence, from above mathematical equations x = π/2

Now, 6 sin-1 (1) = 6 * π/2 = 3 π

Similarly, consider x = sin-1 (1) 

From rearranging, sin x = 1

As we know that, sin π/2 = 1

Here, π/2 = 90°

Hence, from above mathematical equations x = π/2

Now, 8 sin-1(1) = 8 * π/2 = 4 π

Now, the value of 6 sin-1 (1) + 8 sin-1 (1) will be

3 π + 4 π = 7 π

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