Trig Identities #2 Pythagorean Identities

trig 2

Above is a quick refresher on each trig function. You only really need sin, cos, tan – but for convenience:

$tan(\theta ) = \frac{sin(\theta )}{cos(\theta )}$

(think SOHCAHTOA on the unit circle)

$csc(\theta ) = \frac{1}{sin(\theta )}$

$sec(\theta ) = \frac{1}{cos(\theta )}$

$cot(\theta ) = \frac{1}{tan(\theta )} = \frac{cos(\theta )}{sin(\theta )}$

tr2

Recall the formula of a circle is

$x^2+y^2=1$

and if our $x$ and $y$ coordinates $(cos(\theta), sin(\theta))$ lie on the circle (which they do) then:

$cos^2{\theta } + sin^2{\theta } = 1$

Take this identity and divide both sides by $sin^2{\theta }$ :

$\frac{cos^2(\theta )}{sin^2(\theta)} + 1 = \frac{1}{sin^2(\theta)}$

$cot^2(\theta)+1=csc^2(\theta)$

We could instead have divded both sides by $cos^2{\theta}$ :

$1+ \frac{sin^2(\theta )}{cos^2(\theta)} = \frac{1}{cos^2(\theta)}$

$1 + tan^2(\theta) = sec^2(\theta)$

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