In mathematics, sin (short for sine) is a trigonometric function that relates the ratio of the length of the side opposite an angle in a right triangle to the length of the hypotenuse. It is one of the fundamental trigonometric functions and is widely used in various branches of mathematics and science.
The concept of sine can be traced back to ancient civilizations such as the Babylonians and Egyptians, who used similar ratios to solve problems related to triangles. However, it was the Indian mathematicians and astronomers who first introduced the modern definition of sine in the 5th century.
The word "sine" is derived from the Latin word "sinus," which means "bay" or "fold." This term was used by the Indian mathematician Aryabhata to describe the half-chord of a circle. The modern definition of sine as a trigonometric function was later developed by the Persian mathematician Al-Khwarizmi in the 9th century.
The concept of sine is typically introduced in high school mathematics, specifically in trigonometry courses. It is commonly taught in grades 10 or 11, depending on the curriculum. However, some advanced math programs may introduce sine in earlier grades.
The concept of sine involves several key knowledge points, including:
To calculate the sine of an angle, follow these steps:
There are no specific types of sine. However, sine can be positive or negative depending on the quadrant in which the angle lies. In the first and second quadrants, sine is positive, while in the third and fourth quadrants, it is negative.
The sine function possesses several important properties:
To find or calculate the sine of an angle, you can use a scientific calculator or refer to trigonometric tables. However, most calculators have a built-in sine function that directly provides the sine value for a given angle.
The formula for calculating the sine of an angle θ in a right triangle is:
sin(θ) = opposite/hypotenuse
Here, "opposite" refers to the length of the side opposite the angle θ, and "hypotenuse" refers to the length of the hypotenuse of the right triangle.
To apply the sine formula, substitute the values of the opposite side and the hypotenuse into the equation. Then, evaluate the division to obtain the sine value.
For example, if the opposite side is 4 units and the hypotenuse is 5 units, the sine of the angle θ would be:
sin(θ) = 4/5
The symbol or abbreviation for sine is "sin."
The primary method for calculating sine is by using the sine formula or equation mentioned earlier. Additionally, scientific calculators and trigonometric tables provide efficient methods for finding the sine of an angle.
Example 1: Find the sine of an angle θ in a right triangle with an opposite side of length 3 units and a hypotenuse of length 5 units.
Solution: Using the sine formula, sin(θ) = 3/5.
Example 2: Determine the sine of an angle θ in a right triangle with an opposite side of length 7 units and a hypotenuse of length 10 units.
Solution: Applying the sine formula, sin(θ) = 7/10.
Example 3: Calculate the sine of an angle θ in a right triangle with an opposite side of length 12 units and a hypotenuse of length 13 units.
Solution: Using the sine formula, sin(θ) = 12/13.
Question: What is sin? Answer: Sin (sine) is a trigonometric function that relates the ratio of the length of the side opposite an angle in a right triangle to the length of the hypotenuse. It is widely used in mathematics and science to solve problems involving triangles and periodic phenomena.