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FimDinia the alriera sonution 1. Deicrmine the valucs of \( \theta \), for \( -90^{\circ}<0<360^{\circ} \) 1. \( \operatorname{sir}(\theta+100)=0,757 \)

Ask by Erickson Simpson. in South Africa
Mar 17,2025

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The values of \( \theta \) are approximately \(-51^\circ\), \(31^\circ\), and \(309^\circ\).

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To find the values of \( \theta \) that satisfy the equation \( \operatorname{sin}(\theta + 100^\circ) = 0.757 \), we first take the inverse sine (arcsin) of 0.757. This gives us two solutions in the given interval: \( \theta + 100^\circ = \arcsin(0.757) \) and \( \theta + 100^\circ = 180^\circ - \arcsin(0.757) \). After calculating \( \arcsin(0.757) \), we can subtract 100° from both results and adjust them to be within the range of \( -90^\circ < \theta < 360^\circ \). To avoid missing out on alternate solutions, check for periodicity since sine function has a period of 360°. This might introduce additional angles that also satisfy the equation within the desired range. Keep your angle adjustments in check to find all valid \( \theta \) values!

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