Math Doubts

Sin angle sum identity

Expansion form

$\sin{(A+B)}$ $\,=\,$ $\sin{A}\cos{B}$ $+$ $\cos{A}\sin{B}$

Simplified form

$\sin{A}\cos{B}$ $+$ $\cos{A}\sin{B}$ $\,=\,$ $\sin{(A+B)}$

Introduction

The angle sum identity for the sine function is usually called as sin of sum of two angles or sin of compound angle identity.

  1. It is used to expand sin of sum of two angles as the sum of product of sin and cos of both angles.
  2. It is also used to simplify the product of sin and cos of two angles as the sin of sum of two angles.

How to use

The angle sum sine identity is mainly used to expand sine of sum of two angles functions like $\sin{(A+B)}$, $\sin{(x+y)}$, $\sin{(\alpha+\beta)}$ and etc. For example

$(1) \,\,\,\,\,\,$ $\sin{(A+B)}$ $\,=\,$ $\sin{A}\cos{B}$ $+$ $\cos{A}\sin{B}$

$(2) \,\,\,\,\,\,$ $\sin{(x+y)}$ $\,=\,$ $\sin{x}\cos{y}$ $+$ $\cos{x}\sin{y}$

$(3) \,\,\,\,\,\,$ $\sin{(\alpha+\beta)}$ $\,=\,$ $\sin{\alpha}\cos{\beta}$ $+$ $\cos{\alpha}\sin{\beta}$

Proof

Let’s learn how to derive the angle sum sin formula by geometrical approach in trigonometry.

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