HALVED - translation to arabic
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  3. HALVED

HALVED - translation to arabic

TERM IN TRIGONOMETRY
Tan half-angle formula; T-substitution; T substitution; Tangent of halved angle; Tangent half angle formula; T formulae; Weierstrauss Substitution; Tangent half-angle formulas
  • 2}}(''a'' + ''b'')}} are the ratios of the actual distances to the length of the diagonal.

HALVED      

الفعل

قَسَّمَ ; قَسَمَ ; نَصَّفَ ; نَصَّفَ الشَّيْءُ

الصفة

مُنَصَّف

نزل إلى النصف      
halve
قطع الى إثنين      
halve

Definition

Halved
·Impf & ·p.p. of Halve.
II. Halved ·adj Appearing as if one side, or one half, were cut away; dimidiate.
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Wikipedia

Tangent half-angle formula

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:

tan ⁡ 1 2 ( η ± θ ) = tan ⁡ 1 2 η ± tan ⁡ 1 2 θ 1 ∓ tan ⁡ 1 2 η tan ⁡ 1 2 θ = sin ⁡ η ± sin ⁡ θ cos ⁡ η + cos ⁡ θ = − cos ⁡ η − cos ⁡ θ sin ⁡ η ∓ sin ⁡ θ , tan ⁡ 1 2 θ = sin ⁡ θ 1 + cos ⁡ θ = tan ⁡ θ sec ⁡ θ + 1 = 1 csc ⁡ θ + cot ⁡ θ , ( η = 0 ) tan ⁡ 1 2 θ = 1 − cos ⁡ θ sin ⁡ θ = sec ⁡ θ − 1 tan ⁡ θ = csc ⁡ θ − cot ⁡ θ , ( η = 0 ) tan ⁡ 1 2 ( θ ± 1 2 π ) = 1 ± sin ⁡ θ cos ⁡ θ = sec ⁡ θ ± tan ⁡ θ = csc ⁡ θ ± 1 cot ⁡ θ , ( η = 1 2 π ) tan ⁡ 1 2 ( θ ± 1 2 π ) = cos ⁡ θ 1 ∓ sin ⁡ θ = 1 sec ⁡ θ ∓ tan ⁡ θ = cot ⁡ θ csc ⁡ θ ∓ 1 , ( η = 1 2 π ) 1 − tan ⁡ 1 2 θ 1 + tan ⁡ 1 2 θ = ± 1 − sin ⁡ θ 1 + sin ⁡ θ tan ⁡ 1 2 θ = ± 1 − cos ⁡ θ 1 + cos ⁡ θ {\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:

sin ⁡ α = 2 tan ⁡ 1 2 α 1 + tan 2 ⁡ 1 2 α cos ⁡ α = 1 − tan 2 ⁡ 1 2 α 1 + tan 2 ⁡ 1 2 α tan ⁡ α = 2 tan ⁡ 1 2 α 1 − tan 2 ⁡ 1 2 α {\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}}}
https://en.wikipedia.org/wiki/Tangent half-angle formula
Examples of use of HALVED
1. Gazprom also halved supplies last year due to non–payment.
2. It has now almost halved that forecast to $6.4 billion.
3. Over the past two years we‘ve halved our footprint.
4. Severit of symptoms halved After eight weeks, almost 60 per cent of those on the drug said the severity of their symptoms had at least halved.
5. Immigration to the Netherlands has halved since 2000.
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