Sine Angle Sum

The sine of a sum or difference can be rewritten in terms of the sines and cosines of the separate angles. This comes from the geometry of rotations: when you combine two turns, the vertical component of the result depends on both the original sine part and the cosine part of each angle. The minus/plus pattern reflects whether you are adding or subtracting the second angle.

Use this when you need the exact sine of an angle that is built from special angles, such as 15°, 75°, or 105°, especially if those angles have known sine and cosine values. It is also useful in trigonometric proofs and in simplifying expressions involving sums or differences of angles.

• Using a plus sign for both addition and subtraction cases instead of changing the middle sign with the angle operation
• Mixing up sin and cos terms, such as writing sinA sinB or cosA cosB in the formula
• Forgetting to use the exact angle values first, which leads to avoidable rounding errors