Answer: B. [tex]\frac{4}{3}[/tex] .
Step-by-step explanation:
- In a right triangle, the ratios of its sides are called trigonometric ratios.
- The three common trigonometric ratios are the sine (sin), cosine (cos), and tangent (tan).
We have given , [tex]\sin\theta=\dfrac{4}{5}\text{ and }\dfrac{\pi}{2}<\theta<\dfrac{3\pi}{2}[/tex].
Then, [tex]\cos\theta =\sqrt{1-\sin^2\theta}\ [\because\ \sin^2A+\cos^2A=1][/tex]
Put value of [tex]\sin\theta[/tex], we get
[tex]\Rightarrow\ \cos\theta=\sqrt{1-(\frac{4}{5})^2}[/tex]
[tex]\Rightarrow=\sqrt{1-\frac{16}{25}}=\sqrt{\frac{25-16}{25}}=\sqrt{\frac{9}{25}}=\frac{3}{5}[/tex]
Now , as we know , [tex]\tan\theta=\dfrac{\sin\theta}{\cos\theta}=\dfrac{\frac{4}{5}}{\frac{3}{5}}=\frac{4}{3}[/tex]
Hence, the correct option is B. [tex]\frac{4}{3}[/tex] .
