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Sin triple angle formula

Expansion form

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

Simplified form

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

Introduction

It is called sine triple angle formula and used in two different cases.

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

How to use

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

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

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

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

Proof

Learn how to derive the rule of sin triple angle identity by geometrical approach in trigonometry.

Learn Proof
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