# Deriving Sin(a+b) from the Unit Circle: Any Ideas?

• Corky
In summary, the formula 'Sin(a+b)=sinacosb + cosasinb' can be derived from the unit circle by finding the straight line distance between two points with coordinates (cos(a),sin(a)) and (cos(b),sin(b)), and then setting it equal to the straight line distance between (1,0) and the point with coordinates (cos(a-b), sin(a-b)). This will result in the formula 'Sin(a-b)=sinacosb - cosasinb', which can then be modified to get the desired formula.

#### Corky

How do I derive the formula 'Sin(a+b)=sinacosb + cosasinb' from the unit circle. Any ideas would be appreaciated our study group tried and failed.

http://www.clowder.net/hop/cos(a+b).html[/U] [Broken]

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That's hard! (Well, more tedious than hard.)

It's a little easier to prove 'Sin(a-b)=sinacosb - cosasinb'
(and then change the sign on b).

The basic idea is to set up the points whose coordinates are
(cos(a),sin(a)) (i.e. the point a distance a from (0,0) measured along the circle) and (cos(b),sin(b)) and calculate the straight line distance between them (the arc distance, along the circle, is a-b, of course.) Now mark the point whose arc length from (1,0) is also a-b: it's coordinates are (cos(a-b), sin(a-b)) and calculate the straight line distance beween it and (1,0). Since the arclengths are the same, the lengths of these chords are the same. Set the two calculations equal and "grind".

Having actually sat down and done the calculation, I find that my suggestion gives the cos(x+y) and cos(x-y). I'm going to have to think about how to get sin(x+y)!

## 1. What is a unit circle?

A unit circle is a circle with a radius of 1 unit, centered at the origin (0,0) on a Cartesian coordinate plane. It is used in mathematics and physics to understand and solve problems related to angles, trigonometric functions, and complex numbers.

## 2. How is the unit circle related to trigonometry?

The unit circle is closely related to trigonometry, as it provides a visual representation of the six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. The coordinates of points on the unit circle correspond to the values of these functions at different angles.

## 3. Why is the unit circle important?

The unit circle is important because it simplifies calculations involving trigonometric functions, making them easier to visualize and understand. It is also used as a reference point for solving more complex trigonometric problems and for graphing trigonometric functions.

## 4. How do you use the unit circle?

To use the unit circle, you can first label the x-axis with angles in degrees or radians, and then use the coordinates of points on the circle to determine the values of trigonometric functions at those angles. You can also use the unit circle to find the values of inverse trigonometric functions.

## 5. What are some real-world applications of the unit circle?

The unit circle has many real-world applications, including calculating the height of a building or the distance between two points using the Pythagorean theorem, determining the trajectory of a projectile, and analyzing wave patterns and harmonic motion. It is also used in navigation, engineering, and physics.