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# Trigonometry

[ "article:topic", "authorname:ucd7" ]

## Circle

The functions $$\sin (\theta)$$ and $$\cos (\theta)$$ are defined such that, if you drew a circle of radius $$r$$ with a triangle inset like the one below, the lengths of the sides will have the listed values:

As the picture suggests, $$\sin \theta$$ and $$\cos \theta$$ have values that repeat when you increase or decrease $$\theta$$ by increments of $$2 \pi$$.

Using the figure above and the Pythagorean Theorem, we can write

$(r\sin\theta)^2 + (r\cos\theta)^2 = r^2$

Dividing the whole equation by $$r^2$$ we obtain the useful result:

$\sin^2\theta + \cos^2\theta = 1$

where $$\sin^2\theta$$ is just a short way of writing $$(\sin\theta)^2$$.  The above equation is true for any value of $$\theta$$.

Using the above results, we can also derive these equations:

$$\tan\theta = \dfrac{\sin\theta}{\cos\theta}$$

$$\sin\theta = \pm\sqrt{1-\cos^2\theta}$$

$$\cos\theta = \pm\sqrt{1-\sin^2\theta}$$

## Trigonometric Identities

$$\sin A + \sin B = 2\sin\left(\dfrac{A + B}{2}\right)\cos\left(\dfrac{A - B}{2}\right)$$

$$\cos A + \cos B = 2\cos\left(\dfrac{A + B}{2}\right)\cos\left(\dfrac{A - B}{2}\right)$$

$$\sin(A + B) = \sin A \cos B + \sin B \cos A$$

$$\cos(A + B) = \cos A \cos B - \sin A \sin B$$

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

## Small-Angle Approximation

If $$\theta$$, expressed in radians, is close to zero, then we can approximate:

$$\sin\theta \approx \theta$$

$$\cos\theta \approx 1$$

$$\tan\theta \approx \theta$$