Sine of Angle difference formula

Formula

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

It is called sine of angle difference identity. It states that the sin of subtraction of two angles is equal to the subtraction of products of sine and cosine of both angles.

Introduction

Sine functions are often appeared with subtraction of two angles. In order to deal them, sine of difference of two angles identity is derived in trigonometry.

Mathematical form

The sine of difference of two angles formula can be written in several ways, for example $\sin{(A-B)}$, $\sin{(x-y)}$, $\sin{(\alpha-\beta)}$, and so on but it is popularly written in the following three mathematical forms.

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

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

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

Use

The sine of angle difference property is mainly used in two cases.

1. To expand sin of difference of two angles as the subtraction of products of sine and cosine functions.
2. To simplify the subtraction of products of sine and cosine functions as the sine of subtraction of two angles.

Proof

Learn how to derive the angle difference sin identity in mathematical form by geometrical approach in trigonometry.

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