Trigonometry
Angle addition¶
By the way, the opposite is the sine times the hypotenuse, and the adjacent is the cosine times the hypotenuse
In other words, \(opp = sin \cdot hyp\), and \(adj = cos \cdot hyp\).
The black triangle in its bottom-left corner has angle \(\alpha\) and the blue triangle in its bottom-left corner has angle \(\beta\), they are both right triangles, and the length of the top of the blue triangle is 1, and of coarse the combined angle is \(\alpha + \beta\).
The question is: what is the \(sin(\alpha + \beta)\) and \(cos(\alpha + \beta)\)?
I'll start with the \(sin\), which is the length of the purple line, which is the length of the purple line below the upper-black line plus the length of the purple line above the upper-black line.
The length below is the same as the height of the right of the black triangle with hypotenuse \(cos(\beta)\) which is the \(opp\) so \(sin(\alpha)cos(\beta)\).
The length above is in the top right mini blue triangle with hypotenuse \(sin(\beta)\), and I'll leave this as an exercise for the viewer, but the angle in the top-left is \(\alpha\) and it's the \(adj\) so \(cos(\alpha)sin(\beta)\).
In total, \(sin(\alpha + \beta) = cos(\alpha)sin(\beta) + sin(\alpha)cos(\beta)\).
Now with the \(cos\), which is the length of the green line, which is the length of the upper-black line minus the length of the mini blue triangle bottom.
The first length is the same as the length of the bottom of the black triangle with hypotenuse \(cos(\beta)\) which is the \(adj\), so \(cos(\alpha)cos(\beta)\)
the length of second line is in the bottom of the mini blue triangle with hypotenuse \(sin(\beta)\), and as you know, the angle in the top-left is \(\alpha\) and it's the \(opp\) so \(sin(\alpha)sin(\beta)\)
in total, \(cos(\alpha + \beta) = cos(\alpha)cos(\beta) - sin(\alpha)sin(\beta)\)