Matematica — Liceo

How the other formulae are derived

Properties1
tagsmatematica, classe/quarta, topic/formule-goniometriche, concept/formule-di-addizione, concept/archi-associati, function/seno, function/coseno, function/tangente, method/formule-addizione, skill/dimostrare, skill/usare-formule

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Once the formula for cos⁡(α−β)\cos(\alpha-\beta) has been established, all the others descend from it by combining it with the related arcs seen in the previous chapter.

Cosine of a sum. It is enough to write cos⁡(α+β)=cos⁡(α−(−β))\cos(\alpha+\beta) = \cos\bigl(\alpha - (-\beta)\bigr): substituting β→−β\beta \to -\beta and using cos⁡(−β)=cos⁡β\cos(-\beta)=\cos\beta and sin⁡(−β)=−sin⁡β\sin(-\beta)=-\sin\beta, we obtain cos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡β.\cos(\alpha+\beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta.

Sine of a sum. We use the relation sin⁡θ=cos⁡(π2−θ)\sin\theta = \cos\bigl(\tfrac{\pi}{2} - \theta\bigr): sin⁡(α+β)=cos⁡(π2−α−β)=cos⁡((π2−α)−β)=cos⁡(π2−α)cos⁡β+sin⁡(π2−α)sin⁡β\sin(\alpha+\beta) = \cos\Bigl(\tfrac{\pi}{2} - \alpha - \beta\Bigr) = \cos\Bigl(\bigl(\tfrac{\pi}{2}-\alpha\bigr) - \beta\Bigr) = \cos\Bigl(\tfrac{\pi}{2}-\alpha\Bigr)\cos\beta + \sin\Bigl(\tfrac{\pi}{2}-\alpha\Bigr)\sin\beta which, using cos⁡(π2−α)=sin⁡α\cos\bigl(\tfrac{\pi}{2}-\alpha\bigr)=\sin\alpha and sin⁡(π2−α)=cos⁡α\sin\bigl(\tfrac{\pi}{2}-\alpha\bigr)=\cos\alpha, becomes sin⁡αcos⁡β+cos⁡αsin⁡β\sin\alpha\cos\beta+\cos\alpha\sin\beta.

Tangent of a sum. The formula for tan⁡(α+β)\tan(\alpha+\beta) is obtained by dividing sin⁡(α+β)\sin(\alpha+\beta) by cos⁡(α+β)\cos(\alpha+\beta) and then dividing numerator and denominator by cos⁡αcos⁡β\cos\alpha\cos\beta.

Topics: Trigonometric formulae
Concepts: Related arcs · Addition formulae
Functions: Cosine · Sine · Tangent
Methods: Addition formulae
Skills: Proving · Using formulae

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