Cosine to Sine conversion

Formulas

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

A cosine term can be converted in terms of sine in two different possible ways. In one method, square of cosine of an angle can be written in square of sine of same angle. Similarly, cosine of an angle can also be expressed in terms of square of sine of angle but the identity is in square root form.

Proof

The two conversion rules of cosine to sine can be derived in trigonometry by Pythagorean identity. According to Pythagoras trigonometric identity, the sum of squares of sine and cosine of an angle is always one.

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

Move sine squared theta term to right hand side of the equation to express cosine squared theta in terms of sine squared theta terms.

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

To obtain other form, take square root both sides.

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