Sin 1: Concept, Inverse, Conversion

The sine function is one of the three major ratios in trigonometry on which the trigonometric functions and formulas are based. The sine function (sin) of an angle gives the ratio of the perpendicular i.e., the opposite side of the angle to the hypotenuse. In the same way, the inverse sine function (sin^-1) gives the ratio of the hypotenuse to the perpendicular of an angle. In radians, the value of sin 1 is 0.8414709848. The complete trigonometric functions and formulas in trigonometry are based on three primary ratios, namely sine, cosine, and tangent. These trigonometric ratios assist us in determining angles and side lengths in a triangle. In this article, we will discuss sin 1 in depth along with some important questions.

Key Takeaways: Sine function, sine wave, trigonometric ratios, trigonometric functions, unit circle, sin 1.

Also read: Isosceles Triangle Theorems

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What is the Sine Function?

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In trigonometry, we have three main functions. They are the sine, cosine, and tangent functions. A function takes an input, performs a specific operation on it, and returns an output. We all know that trigonometry is mostly concerned with right triangles.

Sin θ = opposite / hypotenuse

For sin (1°),

1° = 1 * π/180 radian = 0.01745°c

Take, x = 1° = 0.01745°c

Sin (x) = x¹/1! – x³/3! + x⁵/5! – x⁷/7! + ……….

Sin (1°) = (0.01745)¹/1! – (0.01745)³/3! + (0.01745)⁵/5! – (0.01745)⁷/7! + ……..

= 0.01745 – 0.00000531/6 + ……………..

= 0.01745 – 0.000000878 + ………………

Sin (1°) = 0.01745

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The Mathematics of a Simple Sine Wave

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A sine wave is something we've all seen before. The squiggly wavy shape found in music equalizers adds clarity to the context. A sine wave is the purest form of audio, representing a specific frequency or total value. In fact, sine waves do not exist in nature in isolation. No natural source of sound produces a single sine wave, but rather a collection of sine waves that are bundled and added together. Surprisingly, any sound can be recreated by simply combining different sine waves.

Although this can become prohibitively expensive and absurdly complicated, there are various synthesis methods for recreating sounds. Sine waves can be thought of as the building blocks of audio, much like Lego. Because sine waves are so important in the study of audio, and because digital systems can easily produce single sine waves, we need to understand where the sine wave comes from. We chose the right-angled triangle for this example because it has a few well-known properties that we can use and leverage. So, apart from the fact that one of its sides forms a 90-degree angle with the other, what do we know about the right triangle? So, since the sum of a triangle's angles is 180 degrees, the sum of the triangle's other two angles must be 90 degrees.

We also know that the Pythagorean Theorem holds true for right triangles, stating that the sum of the squares of the triangle's sides equals the square of the hypotenuse. We know that for right-angled triangles, certain well-defined but arbitrary ratios are conjured up. These ratios, known as trigonometric ratios or functions, provide relationships between the sides and angles of a right-angle triangle. In total, there are six: sine, cos, tan, cosec, sec, and cot. They are just made-up fractions that describe some relationships. These ratios, however, have a variety of practical applications. In this section, we will only look at sine.

Sin θ = opposite/hypotenuse

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What is the Value of Sin 1’s Inverse?

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The inverse sin of one, sin^-1 (1), is a one-of-a-kind value for the inverse sine function. Sin-1(x) returns the angle whose sine is x. As a result, sin^-1 (1) equals the angle whose sine is 1.

We already know that Sin 90 = 1.

Therefore,

90 = sin^-1(1) ( in degrees)

π/2 = sin^-1(1) (in radian)

Because the inverse sin^-1 (1) equals 90° or π/2. The sine function's maximum value is denoted by '1'. As a result, it will occur every 90 degrees, such as at π/2, 3/2, and so on.

As a result of this, we can conclude;

π/2+2 πk = sin^-1(1) (for any integer k)

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Sin (1°) in Trigonometric Functions

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We can represent the sin 1 degrees using trigonometry formulas as:

± √(1-cos²(1°))

± tan 1°/√(1 + tan²(1°))

± 1/√(1 + cot²²(1°))

± √(sec²(1°) - 1)/sec 1°

1/cosec 1°

Note: Because 1° is located in the first quadrant, the final value of sin 1° will be positive.

To represent sin 1°, we can use trigonometric identities such as,

sin(180° - 1°) = sin 179°

-sin(180° + 1°) = -sin 181°

cos(90° - 1°) = cos 89°

-cos(90° + 1°) = -cos 91°

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Sin 1 Degrees Using Unit Circle

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Using the unit circle, find the value of sin 1 degrees: 

Anticlockwise rotate 'r' to form a 1° angle with the positive x-axis. The y-coordinate(0.0175) of the point of intersection (0.9998, 0.0175) of the unit circle and r is equal to the sin of 1 degree.

As a result, sin 1° = y = 0.0175. (approx)

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Are Sin 1 and 1 Sin the Same?

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There are several primary functions in trigonometry that are related to the angles of a right-angled triangle. One of the three primary functions in trigonometry is the sine function. It is defined as the ratio of the largest side to one of the smaller sides of a right-angled triangle adjacent to the 90-degree angle.

Sin 1 denotes that the ratio of the largest side to one adjacent side of a 90-degree angle is one. 1 sin, on the other hand, has no meaning because there is no value written after'sin.' Now we can say that sin 1 has a specific value and that sin 1 is meaningless. As a result, sin 1 and sin 1 are not the same.

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Things to Remember

• A function takes an input, performs a specific operation on it, and returns an output. We all know that trigonometry is mostly concerned with right triangles.
• A sine wave is something we've all seen before. The squiggly wavy shape found in music equalizers adds clarity to the context. 
• A sine wave is the purest form of audio, representing a specific frequency or total value.
• Sin (1°) = 0.01745
• sin^-1(1) = Π/2+2Πk (for any integer k)
• sin 1 and 1 sin are not the same.

Also read: Trapezoid Formula