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championmathgirl

Member

Registered: 2015-06-01

Posts: 20

Trig Functions

Why does cot^2(x) + 1 = csc^2(x) for any x such that x is not an integer multiple of 180 degrees, where cot=cotangent and csc=cosecant?

Cotangent=(cos(x))/(sin(x)) and Cosecant=1/(sin(x))

My understanding is (cos^2(x))/(sin^2(x))+1=1/(sin^2(x)), multiplying the equation by sin^2x to get;

cos^2(x)+sin^2(x)=sin^2(x), but that doesn't get me anywhere with why that's true for all x that's not a multiple of 180. A little help please?

Thanks

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Girls can be just as good as boys at math. We just need to get the same encouragement.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,139

Re: Trig Functions

hi championmathgirl

Let's start with

Why is that true?

In right angled triangle trig, using Pythagoras

Divide by H squared

But we can also define sine cosine and tangent for angles beyond 0-90 by considering a unit circle centred on the origin and letting a point rotate around (0,0) :
defining sin(θ) = y coordinate of the point and cos(θ) as the x coordinate.

http://www.mathsisfun.com/geometry/unit-circle.html

As a result equation (1) continues to hold for all angles.

Now divide by sin^2(θ)

which is the same as

There's a third trig identity like these two which you can derive for yourself as an exercise

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

championmathgirl

Member

Registered: 2015-06-01

Posts: 20

Re: Trig Functions

Thank you so much!

(i think I figured out the third... still working on it though!)

---

Girls can be just as good as boys at math. We just need to get the same encouragement.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,139

Re: Trig Functions

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob