Why is $\sqrt{(\cos^2 \phi + \sin^2\phi)} = 1$?

Question

A rather short question: Why is

$$\sqrt{(\cos^2 \phi + \sin^2\phi)} = 1$$

I have seen that in

Answer 1

Draw a circle with radius one in the plane $xy$ and centre at zero. Its equation is $\sqrt{x^2+y^2} = 1.$ If $\phi$ is an angle and $(x, y)$ are the coordinates of the point making such angle, then $x = \cos \phi$ and $y = \sin \phi,$ hence the relation you wanted.

Answer 2

This follows from a key trigonometric identity: $$\cos^2(x)+\sin^2(x)=1$$ which holds for any $x \in\mathbb R$. See here for more info: https://en.wikipedia.org/wiki/Pythagorean_trigonometric_identity.

It follows from drawing a right-angle triangle with hypotenuse of length $1$. Then the side lengths are $\sin(x)$ and $\cos(x)$, where $x$ is one of the interior angles, so Pythagorean theorem gives the identity.

Answer 3

In a right triangle with sides $a$, $b$, and $c$ (the hypotenuse) opposite to the angles $A$, $B$, and $C$ (the right-angle), the Pythagorean Theorem can be used to reach that identity. (At least this is how I always visualize it.)

$$a^2+b^2 = c^2 \implies \color{blue}{c = \sqrt{a^2+b^2}}$$

$$\sin A = \frac{a}{c} \implies a = c\sin A; \quad \cos A = \frac{b}{c} \implies b = c\cos A$$

$$\implies (c\sin A)^2+(c\cos A)^2 = c^2 \implies c^2\sin^2 A+c^2\cos^2 A = c^2$$

$$\implies c^2(\sin^2 A+\cos^2 A) = c^2 \implies \sin^2 A+\cos^2 A = 1 \implies \color{blue}{\sqrt{\sin^2 A+\cos^2 A} = 1}$$

From which the given identity is derived.