The two identities: tan θ ≡ sin θ ÷ cos θ and sin²θ + cos²θ ≡ 1

By the end of this lesson

By the end you can use $\tan\theta \equiv \dfrac{\sin\theta}{\cos\theta}$ and $\sin^2\theta + \cos^2\theta \equiv 1$ to find the other ratios from one given value (with the right signs), and to simplify expressions.

Definition

Identity

An identity is an equation that is true for every value of the variable for which both sides are defined.

It's written with $\equiv$ ("is identically equal to") instead of $=$ .

Compare: $x + x \equiv 2x$ is true for every $x$ (an identity). But $x + 1 = 5$ is only true for $x = 4$ (an ordinary equation).

$\sin^2\theta$ means $(\sin\theta)^2$ : square the sine, not the angle.

The two you need

$\tan\theta \equiv \frac{\sin\theta}{\cos\theta} \qquad\qquad \sin^2\theta + \cos^2\theta \equiv 1$

Why are they true? On the unit circle, the point $P$ at angle $\theta$ is $(\cos\theta, \sin\theta)$ .

$\tan\theta$ is $\dfrac{\text{opposite}}{\text{adjacent}}$ , which is height $\div$ width $= \dfrac{\sin\theta}{\cos\theta}$ . (For any angle, this is how tan is defined on the unit circle.)
$P$ is on the circle $x^2 + y^2 = 1$ . Put in $x = \cos\theta$ and $y = \sin\theta$ : you get $\cos^2\theta + \sin^2\theta = 1$ . That's Pythagoras.

The second one rearranges two ways, and you'll use both all the time:

$\sin^2\theta \equiv 1 - \cos^2\theta \qquad \cos^2\theta \equiv 1 - \sin^2\theta$

Pythagoras on the unit circle. For any angle,

P = (\cos\theta, \sin\theta)

lies on

x^2 + y^2 = 1

, so the identity holds for every

\theta

, not just acute ones.

Quick check

Multiple choiceQuestion 1 · Recall

Answer the question above to continue.