Sin 1

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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 (π/3) = √3/2 

Sin π = 0

Now using these data, we can write;

sin1=sin[π/3−(π/3−1)]

⟹sin1=sin(π/3)cos(π/3-1)−cosπ/3sin(π/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 ≈ (√3/2×1)−[1/2×(π/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) f'(a)/1! (x-a) + f''(a)/2! (x-a)2 + f3(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 - ⅓! + ⅕! - 1/7! + …..

Or, Sin 1 = 1- ⅙ + 1/120 - 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. 

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, 4sin-11 = 4 * π/2

                      = 2π 

2. Calculate the Value of 2sin1 in Radians.

Solution:

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

Therefore, 2sin1 = 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. 

FAQ (Frequently Asked Questions)

1. Are sin 1 and 1 sin the Same?

Ans: 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?

Ans: 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.