# Conversion Formulas to express sin to cos and cos to sin

The Pythagorean theorem which describes the relation between three sides of a right angled triangle is used in trigonometry to express the mathematical relation between trigonometric functions. The Pythagorean identity of sine and cosine is one of them and can further use to express sine in terms of cosine and vice-versa.

## Formulas

Possibly, trigonometric function sine can be expressed in terms of cosine in two different ways. Similarly, cosine can also be expressed in terms of sine in two different ways but in same pattern. Remember, the following identities in trigonometry are written by considering theta as the angle of a right angled triangle.

1

### Sin in terms of cos

$(1)\,\,\,\,$ $\sin^2 \theta = 1 -\cos^2 \theta$

It is read as sin squared theta is equal to one minus cos squared theta.

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

It is read as sin theta is equal to plus or minus square root of one minus cos squared theta.

Learn the proofs of these two identities to convert sine in terms of cosine.

2

### Cos in terms of sin

$(1)\,\,\,\,$ $\cos^2 \theta = 1 -\sin^2 \theta$

It is read as cos squared theta is equal to one minus sin squared theta.

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

It is read as cos theta is equal to plus or minus square root of one minus sin squared theta.

Similarly, learn the mathematical proofs of these two formulas to express cos terms in terms of sine.