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Concept
Tangent half-angle formula
Summary
In trigonometry, tangent half-angle formulas relate the tangent of half of an angle to trigonometric functions of the entire angle. The tangent of half an angle is the stereographic projection of the circle onto a line. Among these formulas are the following:
From these one can derive identities expressing the sine, cosine, and tangent as functions of tangents of half-angles:
Using double-angle formulae and the Pythagorean identity gives
Taking the quotient of the formulae for sine and cosine yields
Combining the Pythagorean identity with the double-angle formula for the cosine,
rearranging, and taking the square roots yields
and
which, upon division gives
Alternatively,
It turns out that the absolute value signs in these last two formulas may be dropped, regardless of which quadrant α is in. With or without the absolute value bars these formulas do not apply when both the numerator and denominator on the right-hand side are zero.
Also, using the angle addition and subtraction formulae for both the sine and cosine one obtains:
Pairwise addition of the above four formulae yields:
Setting and and substituting yields:
Dividing the sum of sines by the sum of cosines one arrives at:
Applying the formulae derived above to the rhombus figure on the right, it is readily shown that
In the unit circle, application of the above shows that . By similarity of triangles,
It follows that
Weierstrass substitution
In various applications of trigonometry, it is useful to rewrite the trigonometric functions (such as sine and cosine) in terms of rational functions of a new variable . These identities are known collectively as the tangent half-angle formulae because of the definition of . These identities can be useful in calculus for converting rational functions in sine and cosine to functions of t in order to find their antiderivatives.
Geometrically, the construction goes like this: for any point (cos φ, sin φ) on the unit circle, draw the line passing through it and the point (−1, 0). This point crosses the y-axis at some point y = t.
Official source
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