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Cos triple angle formula

Expansion form

$\cos{3\theta} \,=\, 4\cos^3{\theta}-3\cos{\theta}$

Simplified form

$4\cos^3{\theta}-3\cos{\theta} \,=\, \cos{3\theta}$

Introduction

It is called cos triple angle identity and used as a formula in two various cases.

  1. Cos of triple angle is expanded as the subtraction of three times cos of angle from four times cos cubed of angle.
  2. The subtraction of three times cos of angle from four times cos cubed of angle is simplified as cos of triple angle.

How to use

Cosine of triple angle identity is used to either expand or simplify the triple angle cos functions like $\cos{3x}$, $\cos{3A}$, $\cos{3\alpha}$ and etc. For example,

$(1) \,\,\,\,\,\,$ $\cos{3x} \,=\, 4\cos^3{x}-3\cos{x}$

$(2) \,\,\,\,\,\,$ $\cos{3A} \,=\, 4\cos^3{A}-3\cos{A}$

$(3) \,\,\,\,\,\,$ $\cos{3\alpha} \,=\, 4\cos^3{\alpha}-3\cos{\alpha}$

Proof

Learn how to derive the rule of cos triple angle identity by geometric approach in trigonometry.

Learn Proof
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