Expert: Richard J. Raridon - 9/8/2011

Question
Dear Prof Richard

http://en.wikipedia.org/wiki/Trigonometric_functions

Some Formulas
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sin(A + B) = sinAcosB + cosAsinB
cos(A + B) = cosAcosB − sinAsinB
tan(A + B) = tanA + tanB / 1 − tanAtanB


sin 2A = 2 sinAcosA

cos 2A = cos2 A − sin2 A

tan 2A =2 tanA / 1 − tan2 A


sin 3x = sin(2x + x)
= sin 2x cos x + cos 2x sin x
= (2 sin x cos x) cos x + (1 − 2 sin2 x) sin x
= 2 sin x cos2 x + sin x − 2 sin3 x
= 2 sin x(1 − sin2 x) + sin x − 2 sin3 x from the identity cos2 x + sin2 x = 1
= 2 sin x − 2 sin3 x + sin x − 2 sin3 x
= 3 sin x − 4 sin3 x

Similarly to above we can compute cos 3x and tan 3x

Question
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We can compute sin, cosine and tangent values upto what value of nx ? where n = 2,3,4,5 ........?

i.e can we compute values for Sin, Cosine and Tangent for n > 3 ?

For example can we compute

Sin 4x, Sin 7x, Sin 19x ?

Cos 5x, Cos 8x, Cos 20x ?

tan 6x, Sin 9x, tan 21x ?

Thanks & Regards,
Prashant S Akerkar

Answer
First of all, for clarity, you need to write
tan(A+B) = (tanA+tanB)/(1-tan^2A)
tan(2A) = 2tanA/(1-tan^2A)
for any value n beyond 3,
sin(nA) = 2sin(n-1)AcosA - sin(n-2)A
cos(nA) = 2cos(n-1)AcosA - cos(n-2)A
once you have the sine and cosine, you can calculate the tangent