# Angle sum formulas

A trigonometric identity that expresses the relation between a trigonometric function with sum of angles and the trigonometric functions with both angles is called the angle sum trigonometric identity. In trigonometry, the following are some of the angle sum formulas with proofs, uses and problems with solutions.

### Sine angle sum formula

The sine of the sum of two angles is equal to the sum of the products of the sines and cosines of both angles.

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

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

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

### Cosine angle sum formula

The cosine of the sum of two angles is equal to the subtraction of the product of sines of both angles from the product of cosines of both 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 angle sum formula

The tangent of the sum of two angles is equal to the quotient of the sum of the tangents of both angles by the subtraction of the product of tangents of both angles from one.

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

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

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

### Cot angle sum formula

The cotangent of the sum of two angles is equal to the quotient of the subtraction of one from the product of cotangents of both angles by the sum of the cotangents of both angles.

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

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

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

Latest Math Problems

A best free mathematics education website for students, teachers and researchers.

###### Maths Topics

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

###### Maths Problems

Learn how to solve the maths problems in different methods with understandable steps.

Learn solutions

###### Subscribe us

You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites.