To prove the identity sin(a + b) = sin(a) * cos(b) + cos(a) * sin(b), we can start by using the unit circle and some basic trigonometric properties.
Consider two angles a and b represented on the unit circle. The coordinates of a point on the unit circle corresponding to angle a are given by (cos(a), sin(a)), and for angle b, the coordinates are (cos(b), sin(b)).
To find the sine of the sum of these two angles (a + b), we can visualize or use geometry to express this. The sine function corresponds to the y-coordinate in the unit circle, which we need to compute for the angle (a + b).
Using the angle addition formulas, we focus on how these two positions relate. By rotating the unit circle, we can deduce that:
- The x-coordinate, which represents the cosine, is the projection of the angle a plus the projection of angle b.
- The y-coordinate, which represents the sine, combines the respective y-values influenced by the adjacent x-values.
Thus, we arrive at:
sin(a + b) = sin(a) * cos(b) + cos(a) * sin(b)This relationship holds true due to how the angles and their corresponding sides interact geometrically on the unit circle. To further verify this, we can utilize specific angle values and the Pythagorean theorem to confirm the validity across different quadrants.
This identity is fundamental in trigonometry and is widely used in various applications, including physics and engineering.