Solve \( cos \theta= \frac{1}{2} \) in the interval \( 0 ^{\circ} \leqslant \theta \leqslant 360 ^{\circ}\)
\begin{align}
cos^{-1} \left( \frac{1}{2} \right) & = \theta \\
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Principal\ value & = 60 ^{\circ} \quad (by\ calculator) \\
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360^{\circ}-60^{\circ} & = 300^{\circ}\\
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Solutions\ & are \\
\theta & = 60 ^{\circ} \\
\theta & = 300^{\circ}
\end{align}
Solve \( sin \theta= \frac{1}{2} \) in the interval \( 0 ^{\circ} \leqslant \theta \leqslant 180 ^{\circ}\)
\begin{align}
sin^{-1} \left( \frac{1}{2} \right) & = \theta \\
\\
Principal\ value & = 30 ^{\circ} \quad (by\ calculator) \\
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180^{\circ}-30^{\circ} & = 150^{\circ}\\
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Solutions\ & are \\
\theta & = 30 ^{\circ} \\
\theta & = 150^{\circ}
\end{align}
Solve \( tan \theta= -1.5 \) in the interval \( 0 ^{\circ} \leqslant \theta \leqslant 360 ^{\circ}\)
(solutions to 2 d.p.)
\begin{align}
tan^{-1} \left( -1.5 \right) & = \theta \\
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Principal\ value & = - 56.31^{\circ} (to\ 2\ d.p.) \\
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-56.31^{\circ}+180^{\circ} & = 123.69^{\circ}\\
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-56.31^{\circ}+2(180^{\circ}) & = 303.69^{\circ}\\
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Solutions\ in\ & the\ interval\ are \\
\theta & =123.61^{\circ} \\
\theta & = 303.69^{\circ}
\end{align}
Solve \( sin \theta= \frac{\sqrt{3}}{4} \) in the interval \( 5 ^{\circ} \leqslant \theta \leqslant 95 ^{\circ}\)
(solutions to 2 d.p.)
\begin{align}
sin^{-1} \left(\frac{\sqrt{3}}{4} \right) & = \theta \\
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Principal\ value & = 25.66^{\circ} (to\ 2\ d.p.) \\
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180-25.66^{\circ} & = 154.34^{\circ}\\
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Solution\ in\ & the\ interval\ is \\
\theta & = 25.66^{\circ}
\end{align}