Tangent half-angle formula

Trigonometry
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Hipparchus Ptolemy Brahmagupta al-Hasib al-Battani Regiomontanus Viète de Moivre Euler Fourier
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In trigonometry, tangent half-angle formulas relate the tangent of half of an angle to trigonometric functions of the entire angle.^[1]

The tangent of half an angle is the stereographic projection of the circle through the point at angle ${\textstyle \pi }$ radians onto the line through the angles ${\textstyle \pm {\frac {\pi }{2}}}$. Among these formulas are the following:

$${\displaystyle {\begin{aligned}\tan {\tfrac {1}{2}}(\eta \pm \theta )&={\frac {\tan {\tfrac {1}{2}}\eta \pm \tan {\tfrac {1}{2}}\theta }{1\mp \tan {\tfrac {1}{2}}\eta \,\tan {\tfrac {1}{2}}\theta }}={\frac {\sin \eta \pm \sin \theta }{\cos \eta +\cos \theta }}=-{\frac {\cos \eta -\cos \theta }{\sin \eta \mp \sin \theta }},\\[10pt]\tan {\tfrac {1}{2}}\theta &={\frac {\sin \theta }{1+\cos \theta }}={\frac {\tan \theta }{\sec \theta +1}}={\frac {1}{\csc \theta +\cot \theta }},&&(\eta =0)\\[10pt]\tan {\tfrac {1}{2}}\theta &={\frac {1-\cos \theta }{\sin \theta }}={\frac {\sec \theta -1}{\tan \theta }}=\csc \theta -\cot \theta ,&&(\eta =0)\\[10pt]\tan {\tfrac {1}{2}}{\big (}\theta \pm {\tfrac {1}{2}}\pi {\big )}&={\frac {1\pm \sin \theta }{\cos \theta }}=\sec \theta \pm \tan \theta ={\frac {\csc \theta \pm 1}{\cot \theta }},&&{\big (}\eta ={\tfrac {1}{2}}\pi {\big )}\\[10pt]\tan {\tfrac {1}{2}}{\big (}\theta \pm {\tfrac {1}{2}}\pi {\big )}&={\frac {\cos \theta }{1\mp \sin \theta }}={\frac {1}{\sec \theta \mp \tan \theta }}={\frac {\cot \theta }{\csc \theta \mp 1}},&&{\big (}\eta ={\tfrac {1}{2}}\pi {\big )}\\[10pt]{\frac {1-\tan {\tfrac {1}{2}}\theta }{1+\tan {\tfrac {1}{2}}\theta }}&=\pm {\sqrt {\frac {1-\sin \theta }{1+\sin \theta }}}\\[10pt]\tan {\tfrac {1}{2}}\theta &=\pm {\sqrt {\frac {1-\cos \theta }{1+\cos \theta }}}\\[10pt]\end{aligned}}}$$

From these one can derive identities expressing the sine, cosine, and tangent as functions of tangents of half-angles:

$${\displaystyle {\begin{aligned}\sin \alpha &={\frac {2\tan {\tfrac {1}{2}}\alpha }{1+\tan ^{2}{\tfrac {1}{2}}\alpha }}\\[7pt]\cos \alpha &={\frac {1-\tan ^{2}{\tfrac {1}{2}}\alpha }{1+\tan ^{2}{\tfrac {1}{2}}\alpha }}\\[7pt]\tan \alpha &={\frac {2\tan {\tfrac {1}{2}}\alpha }{1-\tan ^{2}{\tfrac {1}{2}}\alpha }}\end{aligned}}}$$

Using double-angle formulae and the Pythagorean identity ${\textstyle 1+\tan ^{2}\alpha =1{\big /}\cos ^{2}\alpha }$ gives

$${\displaystyle \sin \alpha =2\sin {\tfrac {1}{2}}\alpha \cos {\tfrac {1}{2}}\alpha ={\frac {2\sin {\tfrac {1}{2}}\alpha \,\cos {\tfrac {1}{2}}\alpha {\Big /}\cos ^{2}{\tfrac {1}{2}}\alpha }{1+\tan ^{2}{\tfrac {1}{2}}\alpha }}={\frac {2\tan {\tfrac {1}{2}}\alpha }{1+\tan ^{2}{\tfrac {1}{2}}\alpha }},\quad {\text{and}}}$$

$${\displaystyle \cos \alpha =\cos ^{2}{\tfrac {1}{2}}\alpha -\sin ^{2}{\tfrac {1}{2}}\alpha ={\frac {\left(\cos ^{2}{\tfrac {1}{2}}\alpha -\sin ^{2}{\tfrac {1}{2}}\alpha \right){\Big /}\cos ^{2}{\tfrac {1}{2}}\alpha }{1+\tan ^{2}{\tfrac {1}{2}}\alpha }}={\frac {1-\tan ^{2}{\tfrac {1}{2}}\alpha }{1+\tan ^{2}{\tfrac {1}{2}}\alpha }}.}$$

Taking the quotient of the formulae for sine and cosine yields

$${\displaystyle \tan \alpha ={\frac {2\tan {\tfrac {1}{2}}\alpha }{1-\tan ^{2}{\tfrac {1}{2}}\alpha }}.}$$

Combining the Pythagorean identity with the double-angle formula for the cosine, ${\textstyle \cos 2\alpha =\cos ^{2}\alpha -\sin ^{2}\alpha =1-2\sin ^{2}\alpha =2\cos ^{2}\alpha -1,}$

rearranging, and taking the square roots yields

$${\displaystyle \left|\sin \alpha \right|={\sqrt {\frac {1-\cos 2\alpha }{2}}}}$$ and $${\displaystyle \left|\cos \alpha \right|={\sqrt {\frac {1+\cos 2\alpha }{2}}}}$$

which, upon division gives

$${\displaystyle \left|\tan \alpha \right|={\frac {\sqrt {1-\cos 2\alpha }}{\sqrt {1+\cos 2\alpha }}}={\frac {{\sqrt {1-\cos 2\alpha }}{\sqrt {1+\cos 2\alpha }}}{1+\cos 2\alpha }}={\frac {\sqrt {1-\cos ^{2}2\alpha }}{1+\cos 2\alpha }}={\frac {\left|\sin 2\alpha \right|}{1+\cos 2\alpha }}.}$$

Alternatively,

$${\displaystyle \left|\tan \alpha \right|={\frac {\sqrt {1-\cos 2\alpha }}{\sqrt {1+\cos 2\alpha }}}={\frac {1-\cos 2\alpha }{{\sqrt {1+\cos 2\alpha }}{\sqrt {1-\cos 2\alpha }}}}={\frac {1-\cos 2\alpha }{\sqrt {1-\cos ^{2}2\alpha }}}={\frac {1-\cos 2\alpha }{\left|\sin 2\alpha \right|}}.}$$

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:

$${\displaystyle {\begin{aligned}\cos(a+b)&=\cos a\cos b-\sin a\sin b\\\cos(a-b)&=\cos a\cos b+\sin a\sin b\\\sin(a+b)&=\sin a\cos b+\cos a\sin b\\\sin(a-b)&=\sin a\cos b-\cos a\sin b\end{aligned}}}$$

Pairwise addition of the above four formulae yields:

$${\displaystyle {\begin{aligned}&\sin(a+b)+\sin(a-b)\\[5mu]&\quad =\sin a\cos b+\cos a\sin b+\sin a\cos b-\cos a\sin b\\[5mu]&\quad =2\sin a\cos b\\[15mu]&\cos(a+b)+\cos(a-b)\\[5mu]&\quad =\cos a\cos b-\sin a\sin b+\cos a\cos b+\sin a\sin b\\[5mu]&\quad =2\cos a\cos b\end{aligned}}}$$

Setting ${\textstyle a={\tfrac {1}{2}}(p+q)}$ and ${\displaystyle b={\tfrac {1}{2}}(p-q)}$ and substituting yields:

$${\displaystyle {\begin{aligned}&\sin p+\sin q\\[5mu]&\quad =\sin \left({\tfrac {1}{2}}(p+q)+{\tfrac {1}{2}}(p-q)\right)+\sin \left({\tfrac {1}{2}}(p+q)-{\tfrac {1}{2}}(p-q)\right)\\[5mu]&\quad =2\sin {\tfrac {1}{2}}(p+q)\,\cos {\tfrac {1}{2}}(p-q)\\[15mu]&\cos p+\cos q\\[5mu]&\quad =\cos \left({\tfrac {1}{2}}(p+q)+{\tfrac {1}{2}}(p-q)\right)+\cos \left({\tfrac {1}{2}}(p+q)-{\tfrac {1}{2}}(p-q)\right)\\[5mu]&\quad =2\cos {\tfrac {1}{2}}(p+q)\,\cos {\tfrac {1}{2}}(p-q)\end{aligned}}}$$

Dividing the sum of sines by the sum of cosines one arrives at:

$${\displaystyle {\frac {\sin p+\sin q}{\cos p+\cos q}}={\frac {2\sin {\tfrac {1}{2}}(p+q)\,\cos {\tfrac {1}{2}}(p-q)}{2\cos {\tfrac {1}{2}}(p+q)\,\cos {\tfrac {1}{2}}(p-q)}}=\tan {\tfrac {1}{2}}(p+q)}$$

Applying the formulae derived above to the rhombus figure on the right, it is readily shown that

$${\displaystyle \tan {\tfrac {1}{2}}(a+b)={\frac {\sin {\tfrac {1}{2}}(a+b)}{\cos {\tfrac {1}{2}}(a+b)}}={\frac {\sin a+\sin b}{\cos a+\cos b}}.}$$

In the unit circle, application of the above shows that ${\textstyle t=\tan {\tfrac {1}{2}}\varphi }$. By similarity of triangles,

$${\displaystyle {\frac {t}{\sin \varphi }}={\frac {1}{1+\cos \varphi }}.}$$

It follows that

$${\displaystyle t={\frac {\sin \varphi }{1+\cos \varphi }}={\frac {\sin \varphi (1-\cos \varphi )}{(1+\cos \varphi )(1-\cos \varphi )}}={\frac {1-\cos \varphi }{\sin \varphi }}.}$$

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 ${\displaystyle t}$. These identities are known collectively as the tangent half-angle formulae because of the definition of ${\displaystyle t}$. 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. One can show using simple geometry that t = tan(φ/2). The equation for the drawn line is y = (1 + x)t. The equation for the intersection of the line and circle is then a quadratic equation involving t. The two solutions to this equation are (−1, 0) and (cos φ, sin φ). This allows us to write the latter as rational functions of t (solutions are given below).

The parameter t represents the stereographic projection of the point (cos φ, sin φ) onto the y-axis with the center of projection at (−1, 0). Thus, the tangent half-angle formulae give conversions between the stereographic coordinate t on the unit circle and the standard angular coordinate φ.

Then we have

$${\displaystyle {\begin{aligned}&\sin \varphi ={\frac {2t}{1+t^{2}}},&&\cos \varphi ={\frac {1-t^{2}}{1+t^{2}}},\\[8pt]&\tan \varphi ={\frac {2t}{1-t^{2}}}&&\cot \varphi ={\frac {1-t^{2}}{2t}},\\[8pt]&\sec \varphi ={\frac {1+t^{2}}{1-t^{2}}},&&\csc \varphi ={\frac {1+t^{2}}{2t}},\end{aligned}}}$$

and

$${\displaystyle e^{i\varphi }={\frac {1+it}{1-it}},\qquad e^{-i\varphi }={\frac {1-it}{1+it}}.}$$

Both this expression of ${\displaystyle e^{i\varphi }}$ and the expression ${\displaystyle t=\tan(\varphi /2)}$ can be solved for ${\displaystyle \varphi }$. Equating these gives the arctangent in terms of the natural logarithm $${\displaystyle \arctan t={\frac {-i}{2}}\ln {\frac {1+it}{1-it}}.}$$

In calculus, the tangent half-angle substitution is used to find antiderivatives of rational functions of sin φ and cos φ. Differentiating ${\displaystyle t=\tan {\tfrac {1}{2}}\varphi }$ gives $${\displaystyle {\frac {dt}{d\varphi }}={\tfrac {1}{2}}\sec ^{2}{\tfrac {1}{2}}\varphi ={\tfrac {1}{2}}(1+\tan ^{2}{\tfrac {1}{2}}\varphi )={\tfrac {1}{2}}(1+t^{2})}$$ and thus $${\displaystyle d\varphi ={{2\,dt} \over {1+t^{2}}}.}$$

One can play an entirely analogous game with the hyperbolic functions. A point on (the right branch of) a hyperbola is given by (cosh ψ, sinh ψ). Projecting this onto y-axis from the center (−1, 0) gives the following:

$${\displaystyle t=\tanh {\tfrac {1}{2}}\psi ={\frac {\sinh \psi }{\cosh \psi +1}}={\frac {\cosh \psi -1}{\sinh \psi }}}$$

with the identities

$${\displaystyle {\begin{aligned}&\sinh \psi ={\frac {2t}{1-t^{2}}},&&\cosh \psi ={\frac {1+t^{2}}{1-t^{2}}},\\[8pt]&\tanh \psi ={\frac {2t}{1+t^{2}}},&&\coth \psi ={\frac {1+t^{2}}{2t}},\\[8pt]&\operatorname {sech} \,\psi ={\frac {1-t^{2}}{1+t^{2}}},&&\operatorname {csch} \,\psi ={\frac {1-t^{2}}{2t}},\end{aligned}}}$$

and

$${\displaystyle e^{\psi }={\frac {1+t}{1-t}},\qquad e^{-\psi }={\frac {1-t}{1+t}}.}$$

Finding ψ in terms of t leads to following relationship between the inverse hyperbolic tangent ${\displaystyle \operatorname {artanh} }$ and the natural logarithm:

$${\displaystyle 2\operatorname {artanh} t=\ln {\frac {1+t}{1-t}}.}$$

The hyperbolic tangent half-angle substitution in calculus uses $${\displaystyle d\psi ={{2\,dt} \over {1-t^{2}}}\,.}$$

Comparing the hyperbolic identities to the circular ones, one notices that they involve the same functions of t, just permuted. If we identify the parameter t in both cases we arrive at a relationship between the circular functions and the hyperbolic ones. That is, if

$${\displaystyle t=\tan {\tfrac {1}{2}}\varphi =\tanh {\tfrac {1}{2}}\psi }$$

then

$${\displaystyle \varphi =2\arctan {\bigl (}\tanh {\tfrac {1}{2}}\psi \,{\bigr )}\equiv \operatorname {gd} \psi .}$$

where gd(ψ) is the Gudermannian function. The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic ones that does not involve complex numbers. The above descriptions of the tangent half-angle formulae (projection the unit circle and standard hyperbola onto the y-axis) give a geometric interpretation of this function.

Starting with a Pythagorean triangle with side lengths a, b, and c that are positive integers and satisfy a² + b² = c², it follows immediately that each interior angle of the triangle has rational values for sine and cosine, because these are just ratios of side lengths. Thus each of these angles has a rational value for its half-angle tangent, using tan φ/2 = sin φ / (1 + cos φ).

The reverse is also true. If there are two positive angles that sum to 90°, each with a rational half-angle tangent, and the third angle is a right angle then a triangle with these interior angles can be scaled to a Pythagorean triangle. If the third angle is not required to be a right angle, but is the angle that makes the three positive angles sum to 180° then the third angle will necessarily have a rational number for its half-angle tangent when the first two do (using angle addition and subtraction formulas for tangents) and the triangle can be scaled to a Heronian triangle.

Generally, if K is a subfield of the complex numbers then tan φ/2 ∈ K ∪ {∞} implies that {sin φ, cos φ, tan φ, sec φ, csc φ, cot φ} ⊆ K ∪ {∞}.

• List of trigonometric identities
• Half-side formula

• Tangent Of Halved Angle at Planetmath