Trigonometric Addition Formulas

Trigonometric addition formulas are a set of equations that express the trigonometric functions of the sum of two angles in terms of the trigonometric functions of the angles themselves. These formulas are important in both pure and applied mathematics, as they are widely used in fields such as engineering, physics, and astronomy. In this article, we will explore the two most common trigonometric addition formulas: the sum-to-product formulas and the product-to-sum formulas.

Sum-to-Product Formulas

The sum-to-product formulas express the sine, cosine, and tangent of the sum of two angles in terms of the sine, cosine, and tangent of the individual angles. The formulas are as follows:

$\sin(\alpha+\beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$

$\cos(\alpha+\beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$

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

where $\alpha$ and $\beta$ are any two angles.

Example

Let's say we want to find the value of $\cos(15^{\circ})$ without using a calculator. We can use the sum-to-product formula for $\cos(\alpha+\beta)$ and set $\alpha=45^{\circ}$ and $\beta=-30^{\circ}$. Then:

$\cos(45^{\circ}-30^{\circ}) = \cos 45^{\circ}\cos(-30^{\circ}) + \sin 45^{\circ}\sin(-30^{\circ})$

Recall that $\cos(-\theta) = \cos\theta$ and $\sin(-\theta)=-\sin\theta$:

$\cos(15^{\circ}) = \dfrac{\sqrt{2}}{2}\cos 30^{\circ} - \dfrac{\sqrt{2}}{2}\sin 30^{\circ}$

$\cos(15^{\circ}) = \dfrac{\sqrt{2}}{4}+\dfrac{\sqrt{2}}{4}$

$\cos(15^{\circ}) = \dfrac{\sqrt{2}}{2}$

Therefore, $\cos(15^{\circ}) = \dfrac{\sqrt{2}}{2}$.

Product-to-Sum Formulas

The product-to-sum formulas express the sine, cosine, and tangent of the sum of two angles in terms of the sine, cosine, and tangent of the half-angle and double-angle identities. The formulas are as follows:

$\sin\alpha\sin\beta = \dfrac{1}{2}(\cos(\alpha-\beta) - \cos(\alpha+\beta))$

$\cos\alpha\cos\beta = \dfrac{1}{2}(\cos(\alpha-\beta) + \cos(\alpha+\beta))$

$\sin\alpha\cos\beta = \dfrac{1}{2}(\sin(\alpha+\beta) + \sin(\alpha-\beta))$

$\tan\alpha\tan\beta = \dfrac{\cos(\alpha-\beta) - \cos(\alpha+\beta)}{\cos(\alpha+\beta) + \cos(\alpha-\beta)}$

where $\alpha$ and $\beta$ are any two angles.

Example

Let's say we want to find the value of $\cos 75^\circ$ without using a calculator. We can use the product-to-sum formula for $\cos\alpha\cos\beta$ with $\alpha=45^\circ$ and $\beta=30^\circ$. Then:

$\cos 45^\circ\cos 30^\circ = \dfrac{1}{2}(\cos(45^\circ-30^\circ)+\cos(45^\circ+30^\circ))$

$\cos 45^\circ\cos 30^\circ = \dfrac{1}{2}\left(\cos 15^\circ + \dfrac{\sqrt{2}}{2}\right)$

We can then use the sum-to-product formula for $\cos(\alpha+\beta)$ with $\alpha=15^\circ$ and $\beta=30^\circ$ to find $\cos 15^\circ$:

$\cos 15^\circ = \cos(45^\circ-30^\circ) = \cos 45^\circ\cos 30^\circ + \sin 45^\circ\sin 30^\circ$

$\cos 15^\circ = \dfrac{\sqrt{2}}{2}\cos 30^\circ + \dfrac{\sqrt{2}}{2}\sin 30^\circ$

$\cos 15^\circ = \dfrac{\sqrt{2}}{4}+\dfrac{\sqrt{2}}{4}$

$\cos 15^\circ = \dfrac{\sqrt{2}}{2}-\dfrac{1}{2}$

Finally, we can substitute this value into the previous expression to obtain:

$\cos 45^\circ\cos 30^\circ = \dfrac{1}{2}\left(\dfrac{\sqrt{2}}{2}-\dfrac{1}{2}+\dfrac{\sqrt{2}}{2}\right) = \dfrac{\sqrt{2}}{4}$

Therefore, $\cos 75^\circ = \dfrac{\sqrt{2}}{4}$.

Conclusion

Trigonometric addition formulas are powerful tools for simplifying and evaluating expressions involving sine, cosine, and tangent functions. While there are many more formulas and identities involving trigonometric functions, the sum-to-product and product-to-sum formulas are some of the most commonly used ones. With practice and familiarity, they can greatly simplify many problems encountered in mathematics and physics.

Links

三角関数の加法定理[JA]