If (x)/(sinθ)=(y)/(sin(θ-(2 π)/(3)))=(z)/(sin(θ+(2 π)/(3))), then x+y+z is (a) cos2 θ(b) sin2 θ(

step 1 If x upon sine theta equal to y upon sine theta theta negative 2 pi upon 3 equal to z upon sine

If (x)/(sinθ)=(y)/(sin(θ-(2 π)/(3)))=(z)/(sin(θ+(2 π)/(3))),...

If $\frac{\mathrm{x}}{\sin \theta}=\frac{\mathrm{y}}{\sin \left(\theta-\frac{2 \pi}{3}\right)}=\frac{\mathrm{z}}{\sin \left(\theta+\frac{2 \pi}{3}\right)}$, then $\mathrm{x}+\mathrm{y}+\mathrm{z}$ is
(a) $\cos 2 \theta$
(b) $\sin 2 \theta$
(c) 0
(d) $\frac{1}{3}$

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Key Concepts

Sine Rule

The sine rule is a fundamental principle in trigonometry that relates the lengths of the sides of a triangle to the sines of its opposite angles. It states that the ratio of any side length to the sine of its corresponding opposite angle is constant throughout the triangle. This rule is essential in problems where sides and angles are interrelated, allowing for transformations between geometric parameters and trigonometric functions.

Sum-to-Product Identities

Sum-to-product identities are trigonometric formulas used to convert sums or differences of sine or cosine functions into products. These identities help simplify expressions by revealing cancellation or symmetry, making it easier to combine or reduce multiple trigonometric terms, especially when dealing with functions shifted by constant angles.

Phase Shifts in Trigonometric Functions

Phase shifts refer to horizontal translations in the graphs of trigonometric functions and represent adjustments to the input angles by adding or subtracting a constant. Recognizing and handling phase shifts is essential for combining sine or cosine expressions with different arguments, as it provides insight into how these shifted functions interact, potentially resulting in cancellation or reinforcement of terms.

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