Answer:
C. [tex]\sin\theta =-\frac{\sqrt{33}}{7}[/tex] and [tex]\tan\theta=\frac{-\sqrt{33}}{4}}[/tex]
Step-by-step explanation:
We have that, [tex]\cos \theta =\frac{4}{7}[/tex].
As, it is given that, [tex]\sin^{2}\theta +\cos^{2}\theta=1[/tex]
i.e. [tex]\sin^{2}\theta =1-\cos^{2}\theta[/tex]
i.e. [tex]\sin^{2}\theta =1-(\frac{4}{7})^{2}[/tex]
i.e. [tex]\sin^{2}\theta =1-\frac{16}{49}[/tex]
i.e. [tex]\sin^{2}\theta =\frac{49-16}{49}[/tex]
i.e. [tex]\sin^{2}\theta =\frac{33}{49}[/tex]
i.e. [tex]\sin\theta =\pm \frac{\sqrt{33}}{7}[/tex]
Thus, according to options, [tex]\sin\theta =-\frac{\sqrt{33}}{7}[/tex]
Now, as we know that,
[tex]\tan\theta=\frac{\sin\theta}{\cos\theta}[/tex]
i.e. [tex]\tan\theta=\frac{-\frac{\sqrt{33}}{7}}{\frac{4}{7}}[/tex]
i.e. [tex]\tan\theta=\frac{-\sqrt{33}}{4}}[/tex]
Hence, option C is correct.
