The Pythagorean Identity

One of the most fundamental identities in trigonometry comes directly from the Pythagorean theorem. Consider a right-angled triangle with opposite side $o$ , adjacent side $a$ , and hypotenuse $h$ . By the Pythagorean theorem:

$o^{2} + a^{2} = h^{2}$

Dividing every term by $h^{2}$ :

$\frac{o ^{2}}{h ^{2}} + \frac{a ^{2}}{h ^{2}} = 1 \Rightarrow (\frac{o}{h})^{2} + (\frac{a}{h})^{2} = 1$

Since $sin θ = \frac{o}{h}$ and $cos θ = \frac{a}{h}$ , this becomes:

$cos^{2} θ + sin^{2} θ = 1$

Pythagorean Identity: The identity $cos^{2} θ + sin^{2} θ = 1$ , which holds for all real values of $θ$ , not just acute angles in a right triangle.

Note

Although the derivation above uses a right triangle (where $θ$ is acute), the identity $cos^{2} θ + sin^{2} θ = 1$ is valid for any real value of $θ$ . This makes it a powerful tool across all types of trigonometric problems.

Free preview

8 more sections in this topic

The Pythagorean Identity

$o^{2} + a^{2} = h^{2}$

Dividing every term by $h^{2}$ :

$\frac{o ^{2}}{h ^{2}} + \frac{a ^{2}}{h ^{2}} = 1 \Rightarrow (\frac{o}{h})^{2} + (\frac{a}{h})^{2} = 1$

Since $sin θ = \frac{o}{h}$ and $cos θ = \frac{a}{h}$ , this becomes:

$cos^{2} θ + sin^{2} θ = 1$

Pythagorean Identity: The identity $cos^{2} θ + sin^{2} θ = 1$ , which holds for all real values of $θ$ , not just acute angles in a right triangle.

Note

Free preview

SL 3.6—Pythagorean identity, double angles

The Pythagorean Identity

8 more sections in this topic

SL 3.6—Pythagorean identity, double angles

The Pythagorean Identity

8 more sections in this topic