Question

Sin 3a+ sin2a- sin a = 4sin a cos a/2 cos 3a/2

Answer 1

Answer on Question #53993 – Math – Trigonometry

$\sin 3a + \sin 2a - \sin a = 4\sin a\cos a / 2\cos 3a / 2$

Solution

We need to prove

$$\sin 3a + \sin 2a - \sin a = 4\sin a \cos \frac{a}{2} \cos \frac{3a}{2}$$

(1)

It is known that

$$\sin 2a = 2\sin a \cos a,$$ $$\sin 3a = \sin 2a \cos a + \cos 2a \sin a = 2\sin a \cos^2 a + \sin a(1 - 2\sin^2 a) = 3\sin a - 4\sin^3 a,$$ $$\cos \frac{a}{2} \cos \frac{3a}{2} = \frac{1}{2}(\cos a + \cos 2a) = \frac{1}{2} \cos a + \frac{1}{2} - \sin^2 a.$$

Left side of (1): $\sin 3a + \sin 2a - \sin a = 3\sin a - 4\sin^3 a + 2\sin a \cos a - \sin a = 2\sin a - 4\sin^3 a + 2\sin a \cos a$;

Right side of (1): $4\sin a \cos \frac{a}{2} \cos \frac{3a}{2} = 2\sin a \cos a + 2\sin a - 4\sin^3 a$.

So leftside=right side, which proves that (1) holds true for any $a$

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