Table of Contents
Trigonometric ratios of multiple of Angle-3A
Page contents:
proofs of sin 3A, cos 3A, tan 3A, cot 3A
Formulae:
- Sin 3A = 3Sin A-4 $sin^3$ A
- cos 3A = 4 $cos^3$ A - 3cos A
- tan 3A = $\frac{3tan\ A-tan^3\ A}{1-3tan^2\ A}$
- cot 3A = $\frac{3cot\ A-cot^3\ A}{1-3cot^\ A}$
1) Sin 3A = 3Sin A-4 $sin^3$ A
Proof:
Sin 3A = sin(A +2A)
use compund angle formula of sine function, i.e. Sin(A+B)= sinA cosB + cosA sinB
Sin 3A = sin 2A cos A + cos 2A sin A
use Sin 2A = 2sin A cos A; Cos 2A = 1-2 $sin^2 A$
Sin 3A = 2sin A $cos^2$ A + (1-2 $sin^2 A$) sin A
using the trigonometric identity $sin^2$ A + $cos^2$ A =1
sin 3A = 2sin A(1- $sin^2$ A) + sin A - 2 $sin^3 A$
Sin 3A = 3Sin A-4 $sin^3$ A
2) Cos 3A = 4 $cos^3$ A - 3cos A
Proof:
cos 3A = cos(A+2A)
use compund angle formula of sine function, i.e. cos(A+B)= cosA cosB + sinA sinB
cos(A+2A)= cosA cos2A + sinA sin 2A
use Sin 2A = 2sin A cos A; Cos 2A = 2 $cos^2 A$-1
cos 3A = cos A(2 $cos^2 A$-1) + sin A 2sin Acos A
cos 3A = 2 $cos^3 A$ -cos A +2 $sin^2$ A cos A
using the trigonometric identity $sin^2$ A + $cos^2$ A =1
cos 3A = 2 $cos^3 A$ -cos A +2(1- $cos^2$ A) cos A
Cos 3A = 4 $cos^3$ A - 3cos A
3) tan 3A = $\frac{3tan\ A-tan^3\ A}{1-3tan^2\ A}$
Proof:
tan 3A = tan(A+2A)
use compund angle formula of sine function, i.e. tan(A+B) = $\frac{tan\ A+tan\ B}{1-tan\ Atan\ B}$
tan 3A = $\frac{tan\ A+tan\ 2A}{1-tan\ Atan\ 2A}$
use the formula tan 2A = $\frac{2tan\ A}{1-tan^2\ A}$
tan 3A = $\frac{tan\ A+\frac{2tan\ A}{1-tan^2\ A}}{1-tan\ A\frac{2tan\ A}{1-tan^2\ A}}$
tan 3A = $\frac{3tan\ A-tan^3\ A}{1-3tan^2\ A}$
4) cot 3A = $\frac{3cot\ A-cot^3\ A}{1-3cot^\ A}$
Proof:
cot 3A = cot(A+2A)
use compund angle formula of sine function, i.e. cot(A+B) = $\frac{cot\ A cot\ B-1}{cot\ A+cot\ B}$
cot 3A = $\frac{cot\ A cot\ 2A-1}{cot\ A+cot\ 2A}$
use the formula cot 2A = $\frac{cot^2\ A-1}{2cot\ A}$
cot 3A = $\frac{cot\ A \frac{cot^2\ A-1}{2cot\ A}-1}{cot\ A+\frac{cot^2\ A-1}{2cot\ A}}$
cot 3A = $\frac{3cot\ A-cot^3\ A}{1-3cot^\ A}$
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