Sine to Cosine conversion

Formulas

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

A sine term can be expressed in terms of cosine in two forms. In first method, sin squared angle can be written in terms of cos squared angle. Similarly, sin of angle can also be expressed in terms of cos squared angle but through square root form.

Proof

The Pythagorean identity of sine and cosine is used to express the two conversion rules of sine to write it in terms of cos. We know that the sum of squares of sin and cos of an angle is one as per Pythagorean identity.

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

Move cos squared theta term to right hand side of this trigonometric equation to get the expression of square of sin theta in terms of square of cos theta.

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

Take square root both sides to express sin theta in terms of cos squared theta through a square root.

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

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