# Pythagorean identity of Sine and Cosine

## Formula

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

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

It is a trigonometric identity that expresses relation between sine and cosine functions in square form. It is derived by the Pythagorean Theorem. Hence, it is called a Pythagorean trigonometric identity. It is mainly used to express sine in terms of cosine and vice-versa.

### Proof

$\Delta BAC$ is a right angled triangle and its angle is theta.

Express the relation between three sides of the triangle, according to Pythagorean Theorem.

${AC}^2 = {BC}^2 + {AB}^2$

$\implies {BC}^2 + {AB}^2 = {AC}^2$

Divide both sides of the expression by ${AC}^2$.

$\implies \dfrac{{BC}^2 + {AB}^2}{{AC}^2} = \dfrac{{AC}^2}{{AC}^2}$

$\require{cancel} \implies \dfrac{{BC}^2}{{AC}^2} + \dfrac{{AB}^2}{{AC}^2} = \dfrac{\cancel{{AC}^2}}{\cancel{{AC}^2}}$

$\implies \dfrac{{BC}^2}{{AC}^2} + \dfrac{{AB}^2}{{AC}^2} = 1$

$\implies {\Bigg( \dfrac{BC}{AC} \Bigg)}^2 + {\Bigg(\dfrac{AB}{AC}\Bigg)}^2 = 1$

According to $\Delta BAC$

$\dfrac{BC}{AC} = \sin \theta$ and $\dfrac{AB}{AC} = \cos \theta$

Replace ratios of two sides in terms of associated trigonometric functions.

$\implies (\sin \theta)^2 + (\cos \theta)^2 = 1$

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

The rule expresses the mathematical relation between sine and cosine and also represents the Pythagorean theorem in terms of sine and cosine functions.

#### Verification

Assume angle of triangle $\theta = 30^°$ to verify this trigonometric relation.

$\sin^2 \theta + \cos^2 \theta = \sin^2 30^° + \cos^2 30^°$

$\implies \sin^2 30^° + \cos^2 30^° = \Bigg(\dfrac{1}{2}\Bigg)^2 + \Bigg(\dfrac{\sqrt{3}}{2}\Bigg)^2$

$\implies \sin^2 30^° + \cos^2 30^° = \dfrac{1}{4} + \dfrac{3}{4}$

$\implies \sin^2 30^° + \cos^2 30^° = \dfrac{4}{4}$

$\implies \sin^2 30^° + \cos^2 30^° = 1$

It is proved that summation of squares of sine and cosine at angle $30^°$ is equal to $1$. It is also proved for every angle. Hence, it is called as a trigonometric identity.

The Pythagorean identity of sine and cosine is used to derive four additional formulas to express sin in terms of cos and vice-versa.

###### Sine in terms of Cosine

The trigonometric function sin can be expressed in terms of cos in two different types.

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

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

Learn the proof of expressing sin in terms of cos.

###### Cosine in terms of Sine

Similarly, the trigonometric ratio cosine can also be converted in terms of sin in two different ways.

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

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

Learn the proof of deriving cos in terms of sin.

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