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gmany

[tex]\sin^2q+\cos^2q=1\to \cos^2q=1-\sin^2q\to\cos q=\pm\sqrt{1-\sin^2q}\\\\q\ \text{terminates in the fourth quadrant, therefore}\ \cos q > 0.\\\\\sin q=-\dfrac{3}{5}\to\cos q=\sqrt{1-\left(-\dfrac{3}{5}\right)^2}\\\\\cos q=\sqrt{1-\dfrac{9}{25}}\\\\\cos q=\sqrt{\dfrac{25}{25}-\dfrac{9}{25}}\\\\\cos q=\sqrt{\dfrac{16}{25}}\\\\\cos q=\dfrac{4}{5}[/tex]

[tex]\tan q=\dfrac{\sin q}{\cos q}\\\\\sin2q=2\sin q\cos q\\\\\cos2q=\cos^2q-\sin^2q\\\\\text{therefore}\\\\\tan2q=\dfrac{\sin2q}{\cos2q}=\dfrac{2\sin q\cos q}{\cos^2q-\sin^2q}\\\\\text{substitute}\\\\\tan2q=\dfrac{2\left(-\dfrac{3}{5}\right)\left(\dfrac{4}{5}\right)}{\left(\dfrac{4}{5}\right)^2-\left(-\dfrac{3}{5}\right)^2}=\dfrac{-\dfrac{24}{25}}{\dfrac{16}{25}-\dfrac{9}{25}}=\dfrac{-\dfrac{24}{25}}{\dfrac{7}{25}}=-\dfrac{24}{25}\cdot\dfrac{25}{7}=-\dfrac{24}{7}[/tex]

[tex]Answer:\ \boxed{B:\ -\dfrac{24}{7}}[/tex]

The exact value of tan 2θ will be –24/7. Then the correct option is B.

What is trigonometry?

The connection between the lengths and angles of a triangular shape is the subject of trigonometry.

If sin θ = -3/5.

Then the value of cos θ will be

cos² θ = 1 – sin² θ

cos² θ = 1 – (-3/5)²

cos² θ = 1 – 9/25

cos² θ = 16/25

 cos θ = 4/5

Then the value of tan 2θ will be

[tex]\tan 2\theta = \dfrac{\sin 2\theta }{ \cos 2\theta}\\\\\\\tan 2\theta = \dfrac{2 \sin \theta \cos \theta }{ \cos^2 \theta - \sin ^2 \theta}\\\\\\\tan 2\theta = \dfrac{2 * \frac{-3}{5} * \frac{4}{5} }{( \frac{4}{5})^2 - ( \frac{-3}{5}) ^2 }[/tex]

Simplify the equation, we have

tan 2θ = (–24/25) / (7/25)

tan 2θ = –24/7

Then the correct option is B.

More about the trigonometry link is given below.

https://brainly.com/question/22698523

#SPJ2

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