Engineering & Scientific
Complex Cotangent, Secant & Cosecant Formulas (IMCOT, IMSEC & IMCSC)
Evaluate complex reciprocal trigonometric functions cot(z), sec(z), and csc(z) for lossy RF transmission-line stubs and waveguide impedances using exact conjugate denominator rationalization.
Rationalizing 1/sin(z), 1/cos(z), and cos(z)/sin(z) into numeric components
Excel's IMCOT, IMSEC, and IMCSC return text strings such as "0-0.6i" that cannot be multiplied or plotted directly without re-parsing. Separating z = a + b*i into numeric real and imaginary cells via cosh(b) = (EXP(b) + EXP(-b)) / 2 and sinh(b) = (EXP(b) - EXP(-b)) / 2 keeps downstream RF admittance math purely numeric.
Writing sin(z) = u_s + i*v_s with u_s = SIN(a)*cosh(b), v_s = COS(a)*sinh(b) and cos(z) = u_c + i*v_c with u_c = COS(a)*cosh(b), v_c = -SIN(a)*sinh(b), conjugate division yields exact numeric components for csc(z), sec(z), and cot(z).
| Function | Real Component Formula | Imaginary Component Formula | At z = pi/2 + i*ln(2) |
|---|---|---|---|
| IMCSC = 1 / sin(z) | u_s / (u_s^2 + v_s^2) | -v_s / (u_s^2 + v_s^2) | 0.8000 + 0.0000i |
| IMSEC = 1 / cos(z) | u_c / (u_c^2 + v_c^2) | -v_c / (u_c^2 + v_c^2) | 0.0000 + 1.3333i |
| IMCOT = cos(z) / sin(z) | (u_c*u_s + v_c*v_s) / (u_s^2 + v_s^2) | (v_c*u_s - u_c*v_s) / (u_s^2 + v_s^2) | 0.0000 - 0.6000i |
Complex Cotangent Real & Imaginary Components (IMCOT)
Rationalize cos(z) / sin(z) over |sin(z)|^2 = u_s^2 + v_s^2 so stub input reactance stays numeric.
=LET(ang_a, B2, hyp_b, B3, ch_b, (EXP(hyp_b) + EXP(-hyp_b)) / 2, sh_b, (EXP(hyp_b) - EXP(-hyp_b)) / 2, us, SIN(ang_a) * ch_b, vs, COS(ang_a) * sh_b, uc, COS(ang_a) * ch_b, vc, -SIN(ang_a) * sh_b, ROUND((vc * us - uc * vs) / (us ^ 2 + vs ^ 2), 4))Complex Secant & Cosecant Reciprocals (IMSEC & IMCSC)
Invert cos(z) or sin(z) by dividing the complex conjugate by the squared modulus u^2 + v^2.
=LET(ang_a, B2, hyp_b, B3, ch_b, (EXP(hyp_b) + EXP(-hyp_b)) / 2, sh_b, (EXP(hyp_b) - EXP(-hyp_b)) / 2, us, SIN(ang_a) * ch_b, vs, COS(ang_a) * sh_b, ROUND(us / (us ^ 2 + vs ^ 2), 4))Try the example
Use the original values or change them to test your understanding.