# Trigonometric ratios of multiple of Angle-3A

Page contents:

proofs of sin 3A, cos 3A, tan 3A, cot 3A

Formulae:

1. Sin 3A = 3Sin A-4 $sin^3$ A
2. cos 3A = 4 $cos^3$ A - 3cos A
3. tan 3A = $\frac{3tan\ A-tan^3\ A}{1-3tan^2\ A}$
4. 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 = $\frac{3cot\ A-cot^3\ A}{1-3cot^\ A}$