Trigonometric Functions of the Double Arc

Consider an arc of the trigonometric circumference that measures 45°, its double arc is a 90° arc, but this is not means that the value of the trigonometric functions (sine, cosine and tangent) of the double arc is twice that of the arc, by example:
If the arc equals 30°, your double arc will be 60°. Sin 30° = 1/2, sin 60° = √3/2, so we realize that even though 60° is double 30°, sin 60° is not double sin 30°. We can apply this same situation with several other arcs and trigonometric functions, however we will reach the same conclusion.
In general, consider any arc of measure β, its double arc will be 2β, therefore, sin β ≠ sin 2β, that is, sin 2β ≠ 2. sin β.
Thus, to find the value of the trigonometric functions of a double arc (sin 2β, cos 2β and tg 2β) we will have to follow some relationships, between an arc β and its double arc 2β.
These relationships will be made through the trigonometric functions of arc addition. See how:
• Cos 2β
According to the addition of arcs, cos 2β is equal to:

cos 2β = cos (β + β) = cos β. cos β – sin β. sin β
Joining the similar terms we will have:
cos 2β = cos (β + β) = cos² β - sin² β
Therefore, the calculation of cos 2β will be done using the following formula:
cos 2β = cos² β - sin² β
• Sen 2β
According to the addition of arcs, sin 2β is equal to:
Sen 2β = sin (β + β) = sin β. cos β + sin β. cos β
Putting similar terms in evidence we will have:
Sen 2β = sin (β + β) = 2. sin β. cos β
Therefore, the calculation of sin 2β will be done using the following formula:
Sen 2β = 2. sin β. cos β
• tg 2β
According to the addition of arcs, tg 2β is equal to:
tg 2β = tg (β + β) = tg β + tg β
1 - tg x. tg β
Joining the similar terms we will have:
tg 2β = tg (β + β) = 2 tgβ 
1 - tg²β
Therefore, the calculation of tg 2β will be done using the following formula:
tg 2β = 2 tgβ 
1 - tg²β

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by Danielle de Miranda
Graduated in Mathematics
Brazil School Team

Trigonometry - Math - Brazil School

Would you like to reference this text in a school or academic work? Look:

RAMOS, Danielle de Miranda. "Trigonometric functions of the double arc"; Brazil School. Available in: https://brasilescola.uol.com.br/matematica/funcoes-trigonometricas-arco-duplo.htm. Accessed on June 27, 2021.