# What's the formula for cos3A?

lunnatican4

Answered

2021-12-17

What's the formula for ${\mathrm{cos}}^{3}A$?

Answer & Explanation

${\mathrm{cos}}^{3}A$

$=\mathrm{cos}(A+2A)$

$=\mathrm{cos}\left(A\right)\mathrm{cos}\left(2A\right)-\mathrm{sin}\left(A\right)\mathrm{sin}\left(2A\right)$

$=\mathrm{cos}\left(A\right)(2{\mathrm{cos}}^{2}\left(A\right)-1)-\mathrm{sin}\left(A\right)\left(2\mathrm{sin}\left(A\right)\mathrm{cos}\left(A\right)\right)$

$=2{\mathrm{cos}}^{3}\left(A\right)-\mathrm{cos}\left(A\right)-2{\mathrm{sin}}^{2}\left(A\right)\mathrm{cos}\left(A\right)$

$=2{\mathrm{cos}}^{3}\left(A\right)-\mathrm{cos}\left(A\right)-2(1-{\mathrm{cos}}^{2}\left(A\right))\times \mathrm{cos}\left(A\right)$

$=2{\mathrm{cos}}^{3}\left(A\right)-\mathrm{cos}\left(A\right)-2(\mathrm{cos}\left(A\right)-{\mathrm{cos}}^{3}\left(A\right))$

$={\mathrm{cos}}^{3}\left(A\right)-\mathrm{cos}\left(A\right)-2\mathrm{cos}\left(A\right)+{\mathrm{cos}}^{3}\left(A\right)$

$=4{\mathrm{cos}}^{3}\left(A\right)-\mathrm{cos}\left(A\right)-2\mathrm{cos}\left(A\right)$

$=4{\mathrm{cos}}^{3}\left(A\right)-3\mathrm{cos}\left(A\right)$

${\mathrm{cos}}^{3}\left(A\right)=4{\mathrm{cos}}^{3}\left(A\right)-3\mathrm{cos}\left(A\right)$

The formula for ${\mathrm{cos}}^{3}A\text{}\text{is}\text{}4{\mathrm{cos}}^{3}A-3\mathrm{cos}A$

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