Let me copypaste:
"When sin points out vertically to the arch segment's pointer edge, and tan is what the pointer faces at the coordinate X line,
according to school's math, how do you meassure the difference between tan and sin, or the tanX°-sinX°?
Exp.: the difference for 30° = tan30° - sin30°= 0.57735-0.5= 0.07735
Is there an equation for such length?
Let me check whether it is similar to my finding about it. (I'll post after the reply)"
Yes. The equation that will give the difference, D, between the tangent and sine of an angle, x, is:
D(x) = tan(x) - sin(x) [Eq. 1]
If, for some reason you're averse to using sin(x) and remember that tan(x) = sin(x)/cos(x), you can restate that as:
D(x) = tan(x) - sin(x)
= sin(x)/cos(x) - sin(x)
= sin(x)/cos(x) - cos(x)sin(x)/cos(x)
= sin(x)/cos(x) (1 - cos(x))
D = tan(x) (1 - cos(x)) [Eq. 2]
D = tan(x) - tan(x)cos(x) [Eq. 3]
Alternatively, since sin
2(x) = 1 - cos
2(x), you could rewrite it as:
D = tan(x) - (1 - cos
2(x))
1/2 [Eq. 4]
I'm not seeing a big win there, though. Equation 1 is simpler than equations 2, 3, and 4.