Question

Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.(cot θ ―√3) (2 sin θ + √3) = 0

Accepted Answer

Recognize that the equation is a product of two factors equal to zero: $$(\cot 	heta - \sqrt{3})(2 \sin 	heta + \sqrt{3}) = 0. $$According to the zero product property, set each factor equal to zero separately: $$\cot 	heta - \sqrt{3} = 0$$ and $$2 \sin 	heta + \sqrt{3} = 0. $$Solve the first equation $$\cot 	heta - \sqrt{3} = 0 by $$isolating $$\cot 	heta$$: $$\cot 	heta = \sqrt{3}. $$Recall that $$\cot 	heta = \frac{\cos 	heta}{\sin 	heta}$$, and find the angles $$	heta in$$ $$[0^\circ, 360^\circ)$$ where this is true. Solve the second equation $$2 \sin 	heta + \sqrt{3} = 0 by $$isolating $$\sin 	heta$$: $$\sin 	heta = -\frac{\sqrt{3}}{2}. $$Determine the angles $$	heta in$$ $$[0^\circ, 360^\circ)$$ where the sine has this value. For each equation, use the unit circle or known special angles to find all solutions within the interval $$[0^\circ, 360^\circ). $$Remember that $$\cot 	heta = \sqrt{3}$$ corresponds to angles where tangent is $$\frac{1}{\sqrt{3}}$$, and $$\sin 	heta = -\frac{\sqrt{3}}{2}$$ corresponds to specific reference angles in the third and fourth quadrants. Combine all solutions from both equations to write the complete solution set for $$	heta in $$the interval $$[0^\circ, 360^\circ). $$Express answers as exact values or decimal approximations as appropriate.

Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.
(cot θ ―√3) (2 sin θ + √3) = 0

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Concetti chiave

Trigonometric Equations and Zero-Product Property

Cotangent and Sine Functions

Solving Trigonometric Equations on a Specified Interval