The Pythagorean identity and simplifying

Three quick questions from earlier lessons.

The point $(\cos x, \sin x)$ is on the unit circle, whose equation is $X^2 + Y^2 = 1$ . What does that tell you?

An identity is an equation that is true for every value of the variable.

\sin^2 x + \cos^2 x = 1

, so

\cos^2 x = 1 - \sin^2 x

and

\sin^2 x = 1 - \cos^2 x

. Also

\tan x = \dfrac{\sin x}{\cos x}

. Use them to find one value from another (the quadrant gives the sign) and to simplify.

$\cos\theta = -\dfrac{3}{5}$ and $\theta$ is in quadrant III. Find $\sin\theta$ .

\sin^2\theta = 1 - \dfrac{9}{25} = \dfrac{16}{25}

The Pythagorean identity.

\sin\theta = \pm\dfrac{4}{5}

Two square roots are possible.

\sin\theta = -\dfrac{4}{5}

Sine is negative in quadrant III.

Your turn: $\sin\theta = \dfrac{5}{13}$ and $\theta$ is in quadrant I.

\cos^2\theta = \dfrac{144}{169}

1 − 25/169.

\cos\theta = \dfrac{12}{13}

Positive in quadrant I.

$\sin\theta = \frac{1}{5}$ and $\theta$ is in quadrant II. Find the exact value of $\cos\theta$ .
$\sin\theta = \frac{3}{8}$ and $\theta$ is in quadrant II. Find the exact value of $\cos\theta$ .
$\sin\theta = \frac{1}{6}$ and $\theta$ is in quadrant II. Find the exact value of $\cos\theta$ .
$\sin\theta = \frac{1}{3}$ and $\theta$ is in quadrant II. Find the exact value of $\cos\theta$ .

Which expression is equal to $\dfrac{1 - \cos^2 x}{\sin x}$ (where $\sin x \ne 0$ )?
1. $\sin x$
2. $\cos x$
3. $1 - \cos x$

Which equation is true for every $x$ ?
1. $\sin^2 x + \cos^2 x = 1$
2. $\sin x + \cos x = 1$
3. $\sin^2 x - \cos^2 x = 1$
4. $\tan x = \dfrac{\cos x}{\sin x}$

Three questions to check you have got it.

$\sin\theta = \frac{1}{5}$ and $\theta$ is in quadrant II. Find the exact value of $\cos\theta$ .
$\sin\theta = \frac{3}{8}$ and $\theta$ is in quadrant II. Find the exact value of $\cos\theta$ .
$\sin\theta = \frac{1}{6}$ and $\theta$ is in quadrant II. Find the exact value of $\cos\theta$ .

The Pythagorean identity

$\sin^2 x + \cos^2 x = 1$ for every $x$ .
From one value: square, subtract from 1, take the root, choose the sign by quadrant.

Watch out: the identity needs the squares.

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