Inscribed Angle Theorem Tangent Double Angle Tangent of Sum of Angle

Published by Fudgy McFarlen on February 11, 2023February 11, 2023

Research Links

Wikipedia: Inscribed angle
Proof Wiki: Double Angle Formulas/Tangent
PDF: DOUBLE-ANGLE, POWER-REDUCING, AND HALF-ANGLE FORMULAS
Intersecting Chord Theorem for Ellipses – chords and other conical sections

sqrt{(2 + 2 cos theta)}

By inspection from the above diagram:

2theta = arctan( {sin(theta)}/{cos(theta)} ) = 2arctan({sin(theta)} / {1+cos(theta)})

Looking at this with the double angle tangent formula

$tan(2theta) = {2tan(theta)}/{1+tan^2(theta)}$

$y = {2x}/{1+x^2}$ – the above is of this form. Solving for x

yx^2 + 2x - y = 0

$x = { -1 + sqrt(1+y^2)} / {y}$

$tan(theta) = { -1 + sqrt(1+tan(2theta)^2)} / {tan(2theta)}$

tan(theta) = { -1 + 1/(cos(2theta))} / {tan(2theta)}

tan(theta) = { -1 + cos(2theta)} / {sin(2theta)}

Example 1:

Using: $tan(theta) = { -1 + sqrt(1+tan(2theta)^2)} / {tan(2theta)}$ with tan(2theta) = 1

$tan(theta) = { -1 + sqrt(1+tan(2theta)^2)} / {tan(2theta)}$

$tan(theta) = { -1 + sqrt(1+1^2)} / {1} = 0.4142$ so tan(22.5) = 0.4142

Example 2:

Using: atan( {0.3420}/{9397} ) = 19.9999 with sin(20) = 0.3420 cos(20) = 0.9397

Which brings me to questions about the multiplication of vectors

x_2(3,1) + y_2(-1,3) = (5,5)

tan^-1(1/3) = 18.4349

tan^-1(1/2) = 26.565

tan^-1(1/3) + tan^-1(1/2) = 45

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