Given:
The trigonometric ratios are given as,
[tex]\begin{gathered} \tan 143^{\circ} \\ \cos (\frac{\pi}{3}) \\ \sin 362^{\circ} \\ \csc (\frac{3\pi}{4}) \end{gathered}[/tex]
The objective is to find out which of these expressions is positive or negative.
Explanation:
Consider the first expression and convert it as,
[tex]\text{tan}143=\tan (90+53)\text{ . . . .(1)}[/tex]
Using the trigonometric identities,
[tex]\tan (90+\theta)=-\cot \theta[/tex]
The equation (1) can be written as,
[tex]\begin{gathered} \tan (90+53)=-\cot (53\degree)=-\frac{1}{\tan 53\degree} \\ =-0.754 \end{gathered}[/tex]
Thus, tan(143°) is a negative expression.
Consider the second trigonometric expression.
[tex]\begin{gathered} \cos (\frac{\pi}{3})=\cos (\frac{\pi}{3}\times\frac{180}{\pi}) \\ =\cos 60\degree \\ =\frac{1}{2} \end{gathered}[/tex]
Thus, cos (π/3) is a positive expression.
Consider the third trigonometric expression.
[tex]\begin{gathered} \sin 362\degree=\sin (360+2)=\sin 2\degree \\ \sin 2\degree=0.035 \end{gathered}[/tex]
Thus, sin 362° is a positive expression.
Consider the fourth expression.
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