Find The Exact Values Of $\sin 2\theta$, $\cos 2\theta$, And $\tan 2\theta$, And Determine The Quadrant In Which $2\theta$ Lies.Given: $\sin \theta = \frac{45}{53}$, With $\theta$ In Quadrant I.

Introduction

In this article, we will explore the process of finding the exact values of $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$ given the value of $\sin \theta$. We will also determine the quadrant in which $2\theta$ lies. This problem is a classic example of using trigonometric identities to find the values of trigonometric functions for a given angle.

Given Information

We are given that $\sin \theta = \frac{45}{53}$, and $\theta$ lies in quadrant I. This means that $\theta$ is an acute angle, and its sine value is positive.

Recall of Trigonometric Identities

To find the values of $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$, we need to recall the following trigonometric identities:

• $\sin 2\theta = 2\sin \theta \cos \theta$
• $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$
• $\tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta}$

Finding $\cos \theta$

Since we are given the value of $\sin \theta$, we can use the Pythagorean identity to find the value of $\cos \theta$.

$\sin^2 \theta + \cos^2 \theta = 1$

Substituting the given value of $\sin \theta$, we get:

$\left(\frac{45}{53}\right)^2 + \cos^2 \theta = 1$

Simplifying the equation, we get:

$\cos^2 \theta = 1 - \left(\frac{45}{53}\right)^2$

$\cos^2 \theta = 1 - \frac{2025}{2809}$

$\cos^2 \theta = \frac{784}{2809}$

Taking the square root of both sides, we get:

$\cos \theta = \pm \sqrt{\frac{784}{2809}}$

Since $\theta$ lies in quadrant I, $\cos \theta$ is positive.

$\cos \theta = \sqrt{\frac{784}{2809}}$

$\cos \theta = \frac{28}{53}$

Finding $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$

Now that we have the values of $\sin \theta$ and $\cos \theta$, we can find the values of $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$ using the trigonometric identities mentioned earlier.

$\sin 2\theta = 2\sin \theta \cos \theta$

$\sin 2\theta = 2 \left(\frac{45}{53}\right) \left(\frac{28}{53}\right)$

$\sin 2\theta = \frac{1260}{2809}$

$\cos 2\theta = \cos^2 \theta - \sin^2 \theta$

$\cos 2\theta = \left(\frac{28}{53}\right)^2 - \left(\frac{45}{53}\right)^2$

$\cos 2\theta = \frac{784}{2809} - \frac{2025}{2809}$

$\cos 2\theta = -\frac{1241}{2809}$

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

$\tan 2\theta = \frac{\frac{1260}{2809}}{-\frac{1241}{2809}}$

$\tan 2\theta = -\frac{1260}{1241}$

Determination of Quadrant

To determine the quadrant in which $2\theta$ lies, we need to analyze the signs of $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$.

Since $\sin 2\theta$ is positive, $2\theta$ lies in quadrants I or II.

Since $\cos 2\theta$ is negative, $2\theta$ lies in quadrants II or III.

Since $\tan 2\theta$ is negative, $2\theta$ lies in quadrants II or IV.

Combining the above information, we can conclude that $2\theta$ lies in quadrant II.

Conclusion

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Q: What is the process of finding the exact values of $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$?

A: The process of finding the exact values of $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$ involves using trigonometric identities and the given value of $\sin \theta$. We need to recall the following trigonometric identities:

• $\sin 2\theta = 2\sin \theta \cos \theta$
• $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$
• $\tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta}$

We also need to find the value of $\cos \theta$ using the Pythagorean identity.

Q: How do we find the value of $\cos \theta$?

A: We can find the value of $\cos \theta$ using the Pythagorean identity:

$\sin^2 \theta + \cos^2 \theta = 1$

Substituting the given value of $\sin \theta$, we can solve for $\cos \theta$.

Q: What if the value of $\cos \theta$ is negative?

A: If the value of $\cos \theta$ is negative, it means that $\theta$ lies in quadrant II or III. In this case, we need to use the negative value of $\cos \theta$ to find the values of $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$.

Q: How do we determine the quadrant in which $2\theta$ lies?

A: To determine the quadrant in which $2\theta$ lies, we need to analyze the signs of $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$. We can use the following rules:

• If $\sin 2\theta$ is positive, $2\theta$ lies in quadrants I or II.
• If $\cos 2\theta$ is positive, $2\theta$ lies in quadrants I or IV.
• If $\tan 2\theta$ is positive, $2\theta$ lies in quadrants I or III.
• If $\tan 2\theta$ is negative, $2\theta$ lies in quadrants II or IV.

Q: What if the values of $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$ are all negative?

A: If the values of $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$ are all negative, it means that $2\theta$ lies in quadrant III.

Q: Can we use trigonometric identities to find the values of $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$ for any given angle?

A: Yes, we can use trigonometric identities to find the values of $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$ for any given angle. However, we need to make sure that we have the correct values of $\sin \theta$ and $\cos \theta$.

Q: What are some common trigonometric identities that we can use to find the values of $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$?

A: Some common trigonometric identities that we can use to find the values of $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$ include:

• $\sin 2\theta = 2\sin \theta \cos \theta$
• $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$
• $\tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta}$
• $\sin^2 \theta + \cos^2 \theta = 1$

Q: How can we apply trigonometric identities to solve problems involving $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$?

A: We can apply trigonometric identities to solve problems involving $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$ by:

• Using the given values of $\sin \theta$ and $\cos \theta$ to find the values of $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$.
• Applying trigonometric identities to simplify expressions involving $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$.
• Analyzing the signs of $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$ to determine the quadrant in which $2\theta$ lies.

Conclusion

In this Q&A article, we have discussed the process of finding the exact values of $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$ using trigonometric identities. We have also covered common trigonometric identities and how to apply them to solve problems involving $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$.