Answer:
Option B. [tex]tan(2\theta)= -\frac{336}{527}[/tex]
Step-by-step explanation:
we know that
[tex]tan(2\theta)= \frac{sin(2\theta)}{cos(2\theta)}[/tex]
[tex]sin(2\theta)=2(sin(\theta))(cos(\theta))[/tex]
[tex]cos(2\theta)=cos^{2}(\theta)-sin^{2}(\theta)[/tex]
[tex]cos^{2}(\theta)+sin^{2}(\theta)=1[/tex]
we have
[tex]sin(\theta)=\frac{24}{25}[/tex]
step 1
Find the value of cosine of angle theta
[tex]cos^{2}(\theta)+sin^{2}(\theta)=1[/tex]
[tex]cos^{2}(\theta)+(\frac{24}{25})^=1[/tex]
[tex]cos^{2}(\theta)=1-\frac{576}{625}[/tex]
[tex]cos^{2}(\theta)=\frac{49}{625}[/tex]
[tex]cos(\theta)=\frac{7}{25}[/tex]
The value of cosine of angle theta is positive, because angle theta lie on the I Quadrant
step 2
Find [tex]sin(2\theta)[/tex]
[tex]sin(2\theta)=2(sin(\theta))(cos(\theta))[/tex]
we have
[tex]sin(\theta)=\frac{24}{25}[/tex]
[tex]cos(\theta)=\frac{7}{25}[/tex]
substitute
[tex]sin(2\theta)=2(\frac{24}{25})(\frac{7}{25})[/tex]
[tex]sin(2\theta)=\frac{336}{625}[/tex]
step 3
Find [tex]cos(2\theta)[/tex]
[tex]cos(2\theta)=cos^{2}(\theta)-sin^{2}(\theta)[/tex]
we have
[tex]sin(\theta)=\frac{24}{25}[/tex]
[tex]cos(\theta)=\frac{7}{25}[/tex]
substitute
[tex]cos(2\theta)=(\frac{7}{25})^{2}-(\frac{24}{25})^{2}[/tex]
[tex]cos(2\theta)=(\frac{49}{625})-(\frac{576}{625})[/tex]
[tex]cos(2\theta)=-\frac{527}{625}[/tex]
step 4
Find the value of [tex]tan(2\theta)[/tex]
[tex]tan(2\theta)= \frac{sin(2\theta)}{cos(2\theta)}[/tex]
we have
[tex]sin(2\theta)=\frac{336}{625}[/tex]
[tex]cos(2\theta)=-\frac{527}{625}[/tex]
substitute
[tex]tan(2\theta)= -\frac{336}{527}[/tex]
