Trig functions

Domain and Range

Based on Professor Leonard, Trig 8(Precalculus 81)

f(θ)

domain

range

sinθ

[1,1]

cosθ

[1,1]

tanθ

θπ2,3π2,5π2,...

(,)

cscθ

θ0,π,2π,3π...

(,1]  [1,)

secθ

θπ2,3π2,5π2,...

(,1]  [1,)

cotθ

θ0,π,2π,3π...

(,)

Reciprocal Identities

(from Trig 9 precalc 82)

Because (for unit circle) sinθ=y and cscθ=1y, therefore cscθ=1sinθ

Etc. for other recip functions. Easy. Skip

But then here’s equation for circle:

x2+y2=r2

Looks like Pythagorean theorem, because it is! Can always inscribe a right triangle onto a circle this way.

So given: sinθ=1213, find y, r, and x.

So y is 12, r = 13. OR can make it y=1213 with a radius of one! Because with similar triangles, can multiply all our sides by same number to get a similar triangle in same proportion. Can therefore also multiply by 13 to get rid of fraction. For Pythagorean theorem, it’s better to stay with y = 12, r = 13.

So let’s find x (I did) $132=169122=144169144=25=x2x=±5$

Important that ±, because we’re taking a square root here. To know sign, we need two trig functions or quadrant info, because triangle can flip on y axis, putting it into quadrant II, so there – x is negative.

Once again, review, tanθ=yx=sinθcosθ