Question

If sin θ=24/25 and 0 less than or equal to θ less than or equal to π/2, find the exact value of tan 2θ.

Answer 1

Answer:

Option B. $tan(2\theta)= -\frac{336}{527}$

Step-by-step explanation:

we know that

$tan(2\theta)= \frac{sin(2\theta)}{cos(2\theta)}$

$sin(2\theta)=2(sin(\theta))(cos(\theta))$

$cos(2\theta)=cos^{2}(\theta)-sin^{2}(\theta)$

$cos^{2}(\theta)+sin^{2}(\theta)=1$

we have

$sin(\theta)=\frac{24}{25}$

step 1

Find the value of cosine of angle theta

$cos^{2}(\theta)+sin^{2}(\theta)=1$

$cos^{2}(\theta)+(\frac{24}{25})^=1$

$cos^{2}(\theta)=1-\frac{576}{625}$

$cos^{2}(\theta)=\frac{49}{625}$

$cos(\theta)=\frac{7}{25}$

The value of cosine of angle theta is positive, because angle theta lie on the I Quadrant

step 2

Find $sin(2\theta)$

$sin(2\theta)=2(sin(\theta))(cos(\theta))$

we have

$sin(\theta)=\frac{24}{25}$

$cos(\theta)=\frac{7}{25}$

substitute

$sin(2\theta)=2(\frac{24}{25})(\frac{7}{25})$

$sin(2\theta)=\frac{336}{625}$

step 3

Find $cos(2\theta)$

$cos(2\theta)=cos^{2}(\theta)-sin^{2}(\theta)$

we have

$sin(\theta)=\frac{24}{25}$

$cos(\theta)=\frac{7}{25}$

substitute

$cos(2\theta)=(\frac{7}{25})^{2}-(\frac{24}{25})^{2}$

$cos(2\theta)=(\frac{49}{625})-(\frac{576}{625})$

$cos(2\theta)=-\frac{527}{625}$

step 4

Find the value of $tan(2\theta)$

$tan(2\theta)= \frac{sin(2\theta)}{cos(2\theta)}$

we have

$sin(2\theta)=\frac{336}{625}$

$cos(2\theta)=-\frac{527}{625}$

substitute

$tan(2\theta)= -\frac{336}{527}$