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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:

\(\sin\theta \approx \theta\)

\(\cos\theta \approx 1\)

\(\tan\theta \approx \theta\)