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.

right angled triangle

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.

Additional identities

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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