# Sine: Definition & Calculation

• Mgeiss
In summary, the sine function, sin(x), is the ratio of the opposite side to the hypotenuse of a right-angle triangle that has the angle x. It can be calculated using a Taylor series or by tracing a circle on a blackboard. The sine function connects circles and squares, and it is one of many functions that cannot be expressed as a simple equation.

#### Mgeiss

What is Sine, and how do I calculate it?

Draw a right angle triangle, for each angle the ratio of the length of the opposite side to the longest side depends only on the angle. This ratio is called the sine of the angle - it's also written as 'sin'

You don't generally calculate it, you either look up the sin of an angle in a table or use the sin button on your calcualtor.

You can calculate the sine of an angle if you know the side lengths of a right triangle that contains that angle. In that case, the sine of an angle is the ratio of the side across from the angle to the hypotenuse of the triangle. The sine of an angle is well defined so it depends only on the angle and not the triangle. Any triangle that has the angle whose sine you want to calculate will do. You can read more about trigonometry here

An alternate definition that avoids the ambiguity of drawing a triangle where one leg has length 0 is to interpret the sine of an angle t as the ordinate (y-coordinate) of the point on a unit circle centered on a Cartesian plane where a radius at the angle t from the positive real axis intersects the circle. The cosine of t is then the abscissa (x-coordinate) of the same point. This definition contains the right-triangle interpretation and shows naturally the periodicity of the trigonometric functions as well as several identities. Have a look at the diagram here if the description is inadequate.

The sin of an angle is equal to the opposite side divided by the hypotenuse. The triangle should be right and the angle cannot be the 90* one.
You can solve it using a Taylor Series, but it's really long. We learned it in calculus and it's somewhat useless when you have a calculator.

The fact is that "Sine" doesn't mean anything and you can't calculate it. Now the "sine function", sin(x), is what everyone here is talking about and you can approximate it, in the first quadrant, at least, by $$x- x^3/3!+ x^5/5!$$, an abbreviated form of the series Tac-Tics mentioned.

Last edited by a moderator:
sinx^2+cosx^2=1

To summarize the replies, the sine, sin(x) is the ratio of the opposing side to the hypotenuse of a right-angle triangle that has the angle x.

It can't be expressed as any simple equation, but as mentioned, it can be calculated from several kinds of infinite series. (so you can calculate it to any precision you want, depending on how many terms you use)

Welcome to a bigger world of Math! While this is the first one you learn about, there are actually lots and lots of functions like this; ones that can't be expressed as any simple equation. If something has a value such as y = f(x) then it's a function, whether or not you can calculate it!

All the replies seem a bit too complex, so I'll give it a go myself.

Imagine tracing a circle on a blackboard. Each time your hand goes around, it moves left and right and also up and down.

Now, imagine moving your hand up and down in the same manner, but without moving it left and right. Watch what happens: your hand moves quickly when it's near the center, but slows down and reverses direction at the top and bottom.

Your hand is "describing" the sine curve as you move it up and down. The motion begins at the center, and each repetition of the pattern corresponds to going once around the circle. The location of your hand along the up-down axis is the value of the sine function at that point in the repetition.

The sine function is, essentially, the mathematical connection between circles and squares. Think about it: you're made a circle with your arm, but then thought about that circle in terms of a square coordinate system, with two dimensions: up-and-down and left-and-right.

I'll let the rest of these guys teach you the symbols, but that's the idea.

- Warren

sohcahtoa

## What is the definition of sine?

Sine is a trigonometric function that represents the ratio of the opposite side to the hypotenuse of a right triangle.

## How is sine calculated?

Sine can be calculated by dividing the length of the opposite side by the length of the hypotenuse of a right triangle.

## What is the range of sine?

The range of sine is between -1 and 1, inclusive.

## What is the periodicity of sine?

Sine has a periodicity of 2π, meaning that it repeats itself every 2π units on the x-axis.

## What are some real-world applications of sine?

Sine is used in various fields such as engineering, physics, and astronomy for measuring angles, calculating distances, and modeling periodic phenomena.

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