If ωis a complex cube root of unity, then the value of sin[(ω^10+ω^26) π-(π)/(4)] is (a) (-√(3))

If ωis a complex cube root of unity, then the value of...

Key Concepts

Complex Roots of Unity

Complex roots of unity are the solutions to the equation z? = 1 in the complex number system. In the case of cube roots of unity, there are three distinct solutions (one real and two complex) that satisfy the identity 1 + ? + ?² = 0. Understanding these roots is essential in many areas of algebra and number theory, as they exhibit symmetry and cyclic properties that simplify the computation of high powers.

Modular Reduction in Exponentiation

When working with powers of roots of unity, exponents can be reduced modulo n (where n is the degree of the root of unity) because the roots cyclically repeat their values. This concept simplifies expressions by replacing high exponents with their remainders when divided by the order of the unity, enabling easier calculations and simplification in complex expressions.

Trigonometric Function Properties

Trigonometric functions, such as sine, have innate periodic and symmetry properties that facilitate the evaluation of expressions even when dealing with large or negative angles. The sine function is periodic with a period of 2? and exhibits odd symmetry, meaning that sin(-?) = -sin(?). These properties are key to reducing and simplifying trigonometric expressions to more familiar angle measures.

Angle Addition and Subtraction in Trigonometry

The addition and subtraction of angles within trigonometric functions often lead to the use of identities that simplify the evaluation of the function. By breaking down compound angles into sums or differences of known angles, one can leverage standard trigonometric values and properties, which is a fundamental technique in solving trigonometric problems.

Transcript

00:01 In the given question we are told to find the value of sine of the angle sine of the angle omega raised to 10 omega raised to 10 plus omega raised to 26 times pi minus pi by 4 so we have to find the value of this expression right and we are given in the question that omega is the complex cube root of complex cube root of unity, that is complex cube root of one.

00:51 And this means that if you take omega raised to 3, we get the value as 1, right? so let's keep this in mind.

01:00 So one of the properties of the complex cube root of 1, that is omega, is that if we take 1 plus omega plus omega square we get the value of this expression as zero so let's just keep this result in mind and now let's simplify the angle inside the sign expression sign function so we have omega raised to 10 plus omega raised to 26 so we can write this as omega raised to nine times omega plus omega raised to 24 times omega square right so we can write it like this and since we already told that omega raised to three is one we can say that if the power of omega is a number that is divisible by three or we can say if it is a multiple of three then it would have the result equal to one right omega raised to three times n as a power would be equal to one so over here we have omega has a multiple of 3 which is 9.

02:12 So we will have 1 times omega which is equal to omega plus omega raised to 24 where 24 is a multiple of 3.

02:21 So omega raised to 24 is also equal to 1 which means we have 1 times omega square which is omega square.

02:28 Right.

02:29 So now we have omega plus omega square instead of omega raised to 10 plus omega raised to 26 and from this result.

02:39 We can rewrite this result and write omega plus omega square is then equal to minus 1.

02:49 So let's make this substitution in the sign function...

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