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sin(teta) = -√3/2

pi<teta<3pi/2

because of range of pi we can conclude that teta is in third quadrant of x-y coordinate system which will determine us what is the sign of cos(teta) and tan(teta)
cos(teta) = √(1-sin^2(teta)) = 1/2
but because teta is in third quadrant, cos(teta) has to be negative which means:
cos(teta) = -1/2

tan(teta) = sin(teta)/cos(teta) = √3

tan of an angle that is in third quadrant is positive which is what we got.

Answer is:
cos(teta) = -1/2
tan(teta) = √3

Answer:

The values of cos Θ and tan Θ are [tex]-\frac{1}{2}[/tex] and √3 respectively.

Step-by-step explanation:

Given,

[tex]sin\theta = -\frac{\sqrt{3} }{2}[/tex]

Since, by the trigonometric identity,

[tex]sin^2 \theta + cos^2\theta = 1[/tex]

[tex]\implies cos^2 \theta = 1 - sin^2 \theta[/tex]

[tex]\implies cos \theta = \pm \sqrt{1 - sin^2 \theta}[/tex]

We have,

[tex]\pi < \theta < \frac{3\pi}{2}[/tex]

⇒ [tex]\theta[/tex] lies in third quadrant,

⇒ [tex]cos\theta [/tex] is negative.

[tex]\implies cos \theta = - \sqrt{1 - sin^2 \theta}=-\sqrt{1 -(-\frac{\sqrt{3} }{2})^2}=-\sqrt{1 -\frac{3}{4}}=-\sqrt{\frac{1}{4}}=-\frac{1}{2}[/tex]

Now,

[tex]tan \theta = \frac{sin \theta}{cos \theta}=\frac{-\sqrt{3}/2}{-1/2}=\sqrt{3}[/tex]

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