Proving the Equation: sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
To prove the identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b), we will use the unit circle and properties of sine and cosine functions.
1. Understanding the Unit Circle
The unit circle is a circle with a radius of 1, centered at the origin (0,0) of a coordinate plane. The x-coordinate of any point on the unit circle is the cosine of the angle formed with the positive x-axis, and the y-coordinate is the sine of that angle. Therefore:
x = cos(θ), y = sin(θ)2. Visualizing the Angles
Consider two angles, a and b. When plotted on the unit circle, the coordinates of the point for angle a are (cos(a), sin(a)), and for angle b they are (cos(b), sin(b)).
3. Using the Angle Sum
When we add these two angles, we get a new angle (a + b). The coordinates of this resulting point on the unit circle will be:
(cos(a + b), sin(a + b))4. Expanding Using Coordinates
To find sin(a + b), we need to express it using the known coordinates of cos(a) and cos(b). We can achieve this by rotating the vectors:
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)5. Using Algebra
To derive this identity, we will use the definitions of sine and cosine from the angles a and b:
- Start with:
sin(a + b) = sin(a)cos(b) + cos(a)sin(b) - Substituting Using Trigonometric Values:
– Substitute the sine and cosine values using their respective coordinates from the unit circle.
6. Conclusion
Thus, by examining the coordinates on the unit circle and leveraging the properties of sine and cosine, we can successfully prove that:
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)This identity plays a crucial role in various applications throughout mathematics and physics, forming the foundation for more complex trigonometric manipulations.