Pythagorean Trigonometric identity of Sine and Cosine

sine squared angle plus cosine squared angle is equal to 1

Definition

Addition of squares of sine and cosine at an angle equals to 1 is called Pythagorean trigonometric identity of sine and cosine.

The relation of sine with cosine is developed by the Pythagorean Theorem. Hence, it is called as Pythagorean trigonometric identity and also called as Pythagoras trigonometric identity in trigonometry. It is mainly used to express trigonometric function sine in terms of cosine and also cosine in terms of sine.

Proof

Consider a right angled triangle $BAC$ and it is assumed that angle of the triangle is theta $\theta$.

right angled triangle

According to Pythagorean Theorem,

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

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

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

$$\Rightarrow \frac{{BC}^2 + {AB}^2}{{AC}^2} = \frac{{AC}^2}{{AC}^2}$$

$$\Rightarrow \frac{{BC}^2}{{AC}^2} + \frac{{AB}^2}{{AC}^2} = \frac{{AC}^2}{{AC}^2}$$

$$\Rightarrow \frac{{BC}^2}{{AC}^2} + \frac{{AB}^2}{{AC}^2} = 1$$

$$\Rightarrow {\Bigg( \frac{BC}{AC} \Bigg)}^2 + {\Bigg(\frac{AB}{AC}\Bigg)}^2 = 1$$

According to $\Delta BAC$

$$\frac{BC}{AC} = \sin \theta$$

$$\frac{AB}{AC} = \cos \theta$$

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

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

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

Sine squared angle plus cosine squared angle is equal to one. It is derived from Pythagorean Theorem. Therefore, it is called as a trigonometric Pythagoras identity.

Verification

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

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

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

$$\Rightarrow \sin^2 30^° + \cos^2 30^° = \frac{1}{4} + \frac{3}{4}$$

$$\Rightarrow \sin^2 30^° + \cos^2 30^° = \frac{4}{4}$$

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

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 trigonometric identity of sine and cosine can also be used to express sine in terms of cosine and cosine in terms of sine.

1
Sine in terms of Cosine

Sine can be expressed in terms of cosine in two different forms.

$\sin^2 \theta = 1 \, – \, \cos^2 \theta$

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

2
Cosine in terms of Sine

Cosine can also be expressed in terms of sine in two different forms.

$\cos^2 \theta = 1 \, – \, \sin^2 \theta$

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

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