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Tanzania [10]
3 years ago
14

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.

Mathematics
1 answer:
GuDViN [60]3 years ago
6 0

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}

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