Math Doubts

Angle sum formulas

The trigonometric functions appear with compound angles, which are formed by the sum of two or more angles. The value of each trigonometric function which contains a compound angle can be expanded in terms of trigonometric functions of angles. They are used as formulas in trigonometry and actually formed by the sum of the angles. Hence, the trigonometric identities are called as angle sum identities.

There are four popularly used angle sum identities. Every angle sum identity is written in three different ways but the formula is same.

Sin of sum of two angles

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

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

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

Cos of sum of two angles

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

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

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

Tan of sum of two angles

$(1) \,\,\,\,\,\,$ $\tan{(A+B)}$ $\,=\,$ $\dfrac{\tan{A}+\tan{B}}{1-\tan{A}\tan{B}}$

$(2) \,\,\,\,\,\,$ $\tan{(x+y)}$ $\,=\,$ $\dfrac{\tan{x}+\tan{y}}{1-\tan{x}\tan{y}}$

$(3) \,\,\,\,\,\,$ $\tan{(\alpha+\beta)}$ $\,=\,$ $\dfrac{\tan{\alpha}+\tan{\beta}}{1-\tan{\alpha}\tan{\beta}}$

Cot of sum of two angles

$(1) \,\,\,\,\,\,$ $\cot{(A+B)}$ $\,=\,$ $\dfrac{\cot{B}+\cot{A}}{\cot{B}\cot{A}-1}$

$(2) \,\,\,\,\,\,$ $\cot{(x+y)}$ $\,=\,$ $\dfrac{\cot{y}+\cot{x}}{\cot{y}\cot{x}-1}$

$(3) \,\,\,\,\,\,$ $\cot{(\alpha+\beta)}$ $\,=\,$ $\dfrac{\cot{\beta}+\cot{\alpha}}{\cot{\beta}\cot{\alpha}-1}$

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