Trigonometry may be the branch of mathematics that considers the relationships between lengths of sides and angles but its mathematics is almost entirely about the equality or otherwise of numerical values.
In a right angle triangle the Sine of an angle is the ratio between the side Opposite the angle and its Hypotenuse and the Cosine of that angle is the ratio of the side Adjacent to the angle and the Hypotenuse. Remember SOH CAH, your chinese friend who lives at TOA where Tangents are the ratio of Opposites and Adjacents.
In our diagram,
cosθ = x/r, sinθ = y/r, tan θ = y/x
All are the ratios of lengths and therefore numbers so when you encounter sin, cos and tan in pure maths you are handling numbers not angles or lengths.
x and y are numerical lengths that lead to coordinate points on our circle. They are always at right angles and Pythagoras applies so
x2 + y2 = r2 .
It is the equation of a circle of radius r which could of course be written as
x2/ r2 + y2/ r2 = 1 and therefore as cos 2 θ + sin 2 θ = 1
Get your calculator out and put this to the test. It is purely a numerical equality.
The following are trig equalities
sin(A ± B) = sinAcosB ± cosAsinB
cos( A + B) = cosAcosB – sinAsinB
cos( A – B) = cosAcosB + sinAsinB
Here is just one proof
The blue length value is sin a
The red length value is cos a
Together the yellow and green length values are sin(a + b)
But in the top triangle cos b = yellow/blue
and in the lowest triangle sin b = green/red
which makes yellow = cos b sin a (as blue = sin a)
and green = sin b cos a ( as red = cos a)
So sin (a + b) (yellow + green ) = cos b sin a + sin b cos a
Again try it on your calculator. It is an identity and always true.
In any triangle like that shown the cosine rule says c2 = a2 + b2 – 2abcosC.
Lets prove it using pythagoras
c2 = (b – acosC) 2 + (asinC) 2
= b 2 + a 2 (cos 2 C + sin 2 C) – 2abcosC
As proved above cos 2 C + sin 2 C = 1
So c 2 = a 2 + b 2 – 2abcosC The cosine rule