Deriving trigonometric identities without the use of unit circles

It is entirely possible to derive trigonometric identities without referring to unit circles. This feat can easily be achieved through the use of SOH CAH TOA, Pythagoras’s theorem (including a right angled triangle diagram) and basic algebra.

Let me demonstrate how this is possible…

Firstly, let’s draw a right angled triangle as such and label its parts…

Right angled triangle

Next let’s write down the fractions which are related to SOH CAH TOA…

\sin { \theta =\frac { O }{ H } } \\ \\ \cos { \theta =\frac { A }{ H } } \\ \\ \tan { \theta =\frac { O }{ A } }

We know that the letter “O” stands for “Opposite”, whilst the letter “H” stands for “Hypotenuse” and the letter “A” stands for “Adjacent”… These are all lengths. Now using Pythagoras’s theorem we should be able to derive the formula below:

{ A }^{ 2 }+{ O }^{ 2 }={ H }^{ 2 }

It turns out that the formula above can be manipulated as such to form (sinθ)^2+(cosθ)^2=1:

\frac { { A }^{ 2 } }{ { H }^{ 2 } } +\frac { { O }^{ 2 } }{ { H }^{ 2 } } =\frac { { H }^{ 2 } }{ { H }^{ 2 } } \\ \\ \therefore \quad { \left( \frac { A }{ H } \right) }^{ 2 }+{ \left( \frac { O }{ H } \right) }^{ 2 }=1\\ \\ \therefore \quad \cos ^{ 2 }{ \theta } +\sin ^{ 2 }{ \theta =1 }

So, what we have is our first trigonometric identity which was derived quite easily.

If one follows the same rules of logic he will be able to derive other similar trigonometric identities.

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