One θ \theta , Two θ \theta , Three θ \theta , n θ n\theta

Geometry Level 2

Which of the following is an expression of sin ⁡ ( n θ ) \sin (n\theta) , where n n is any real number?

n sin ⁡ ( 1 n θ ) cos ⁡ ( 1 n θ ) n\sin(\frac{1}{n}\theta) \cos(\frac{1}{n}\theta) n sin ⁡ ( 1 2 θ ) cos ⁡ ( 1 2 θ ) n\sin(\frac{1}{2}\theta) \cos(\frac{1}{2}\theta) 2 sin ⁡ ( 1 n θ ) cos ⁡ ( 1 n θ ) 2\sin(\frac{1}{n}\theta) \cos(\frac{1}{n}\theta) 2 sin ⁡ ( n θ ) cos ⁡ ( 1 2 n θ ) 2\sin (n\theta) \cos(\frac{1}{2}n\theta) 2 sin ⁡ ( 1 2 n θ ) cos ⁡ ( 1 2 n θ ) 2\sin(\frac{1}{2}n\theta) \cos(\frac{1}{2}n\theta) 2 sin ⁡ ( 1 2 n θ ) cos ⁡ ( n θ ) 2\sin(\frac{1}{2}n\theta) \cos (n\theta)

Andrei Li by Andrei Li , 16, Canada

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1 solution

sin ⁡ α + sin ⁡ β = 2 sin ⁡ α + β 2 cos ⁡ α − β 2 \sin\alpha + \sin\beta=2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha -\beta}{2}

Let be α = n θ \alpha=n\theta and β = 0 \beta=0

sin ⁡ ( n θ ) + sin ⁡ ( 0 ) = 2 sin ⁡ n θ + 0 2 cos ⁡ n θ − 0 2 = 2 sin ⁡ n θ 2 cos ⁡ n θ 2 = sin ⁡ ( n θ ) \sin(n\theta) + \sin(0)=2\sin\frac{n\theta+0}{2}\cos\frac{n\theta -0}{2}=2\sin\frac{n\theta}{2}\cos\frac{n\theta}{2}=\sin(n\theta)

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