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Popular >

simplify (e^{iθ}-e^{-iθ})^6

Solution

simplify

(e^{i θ} - e^{- i θ})^{6}

Solution

- 64 sin^{6} (θ)

Solution steps

(e^{i θ} - e^{- i θ})^{6}

e^{i θ} = cos (θ) + i sin (θ)

e^{- i θ} = cos (θ) - i sin (θ)

= (cos (θ) + i sin (θ) - (cos (θ) - i sin (θ)))^{6}

Expand

cos (θ) + sin (θ) i - (cos (θ) - i sin (θ)) : 2 i sin (θ)

= (2 i sin (θ))^{6}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{6} i^{6} sin^{6} (θ)

2^{6} = 64

= 64 i^{6} sin^{6} (θ)

i^{6} = - 1

= 64 (- 1) sin^{6} (θ)

Refine

= - 64 sin^{6} (θ)

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Frequently Asked Questions (FAQ)

What is (e^{iθ}-e^{-iθ})^6 ?
The solution to (e^{iθ}-e^{-iθ})^6 is -64sin^6(θ)