# Addition and Subtraction Formulas for Sine and Cosine

In a right triangle with legs a and b and hypotenuse c, and angle α opposite side a, the trigonometric functions *sine* and *cosine* are defined as

sinα = a/c, cosα = b/c.

This definition only covers the case of acute positive angles α:

sin(90° - α) = cosα and cos(90° - α) = sinα,

for 0 < α < 90° because these limitations of α also imply the same limitations on

We are concerned here with illustrating two pairs of formulas known as the *Sine and Cosine Addition and Subtraction* formulas, i.e., the formulas for sin(α±β) and cos(α±β), where all the angles involved satisfy the basic limitations:

0 < α, β, α + β, < 90°, for addition, and | |

0 < α, β, α - β < 90°, for subtraction. |

Naturally, after the common extension of the definitions, the formulas remain true for all values of the two angles.

So, given two right triangles, one with angle α and the other with angle β. For the sake of a geometric illustration, we need to put those triangles somehow together to make a combination of angles α and β to stand out. As is suggested in [Gelfand & Saul, p. 126], there are just three ways of doing that:

The third one is the basis for the derivation of the formulas for

### Formulas for sin(α + β) and cos(α + β)

sin(α + β) = sinα cos β + cos α sin β. | |

cos(α + β) = cosα cos β - sin α sin β. |

### Formulas for sin(α - β) and cos(α - β)

sin(α - β) = sinα cos β - cos α sin β. | |

cos(α - β) = cosα cos β + sin α sin β. |

### Another Proof of sin(α + β) = sinα cos β + cos α sin β

The next proof relies on the diagram:

From the definition of *sine* and the fact that the area of a triangle is half the product of the altitude and the base, it follows that the are of a triangle is half the product of any of the two side times sin of the included angle. For the three triangles in the diagram we have

2A_{c} | = c (d cosβ) sinα, | |

2A_{d} | = d (c cosα) sinβ, | |

2A | = cd sin(α + β). |

Which after cancelling cd gives the addition formula for sine.

Observe that in this proof α + β need not be acute. (This proof alongside a similar one for

### References

- I. M. Gelfand, M. Saul,
*Trigonometry*, Birkhäuser, 2001 - R. B. Nelsen,
*Proofs Without Words*, MAA, 1993 - R. B. Nelsen,
*Proofs Without Words II*, MAA, 2000 - J. Zimba,
__On the Possibility of Trigonometric Proofs of the Pythagorean Theorem__,*Forum Geometricorum*, Volume 9 (2009) 275-278

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