Given Cos ⁡ Θ = 3 5 \cos \theta=\frac{3}{5} Cos Θ = 5 3 ​ And Θ \theta Θ Lies In Quadrant IV, Match Each Double-angle Trigonometric Expression With The Corresponding Exact Value.$[ \begin{array}{|l|l|} \hline \text{Double-Angle Trigonometric Expression} &

Introduction

In trigonometry, double-angle trigonometric expressions are used to find the values of trigonometric functions for double angles. These expressions are essential in solving various trigonometric problems, especially in calculus and engineering applications. In this article, we will focus on solving double-angle trigonometric expressions in Quadrant IV, given that $\cos \theta=\frac{3}{5}$ and $\theta$ lies in Quadrant IV.

Understanding Quadrant IV

Quadrant IV is one of the four quadrants in the Cartesian coordinate system. It is located in the upper-left region of the coordinate plane, where both the x-coordinate and y-coordinate are positive. In trigonometry, Quadrant IV is associated with angles that lie between $90^\circ$ and $180^\circ$. Since $\theta$ lies in Quadrant IV, we can expect the values of sine and cosine to be positive.

Recall of Double-Angle Trigonometric Identities

Before we proceed to solve the double-angle trigonometric expressions, let's recall the double-angle trigonometric identities:

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

Solving Double-Angle Trigonometric Expressions

Given that $\cos \theta=\frac{3}{5}$ and $\theta$ lies in Quadrant IV, we can use the double-angle trigonometric identities to find the exact values of the double-angle trigonometric expressions.

Solving $\sin 2\theta$

To solve $\sin 2\theta$, we can use the double-angle trigonometric identity:

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

Since $\cos \theta=\frac{3}{5}$, we can substitute this value into the equation:

$\sin 2\theta = 2\sin \theta \cdot \frac{3}{5}$

However, we still need to find the value of $\sin \theta$. Since $\theta$ lies in Quadrant IV, we can use the Pythagorean identity:

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

Substituting the value of $\cos \theta$, we get:

$\sin^2 \theta + \left(\frac{3}{5}\right)^2 = 1$

Simplifying the equation, we get:

$\sin^2 \theta = 1 - \frac{9}{25} = \frac{16}{25}$

Taking the square root of both sides, we get:

$\sin \theta = \pm \sqrt{\frac{16}{25}} = \pm \frac{4}{5}$

Since $\theta$ lies in Quadrant IV, the value of $\sin \theta$ is positive. Therefore, we can substitute the value of $\sin \theta$ into the equation:

$\sin 2\theta = 2 \cdot \frac{4}{5} \cdot \frac{3}{5} = \frac{24}{25}$

Solving $\cos 2\theta$

To solve $\cos 2\theta$, we can use the double-angle trigonometric identity:

$\cos 2\theta = 2\cos^2 \theta - 1$

Substituting the value of $\cos \theta$, we get:

$\cos 2\theta = 2 \cdot \left(\frac{3}{5}\right)^2 - 1$

Simplifying the equation, we get:

$\cos 2\theta = 2 \cdot \frac{9}{25} - 1 = \frac{18}{25} - \frac{25}{25} = -\frac{7}{25}$

Solving $\tan 2\theta$

To solve $\tan 2\theta$, we can use the double-angle trigonometric identity:

$\tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta}$

However, we still need to find the value of $\tan \theta$. Since $\cos \theta=\frac{3}{5}$, we can use the Pythagorean identity:

$\tan^2 \theta + 1 = \sec^2 \theta$

Substituting the value of $\cos \theta$, we get:

$\tan^2 \theta + 1 = \frac{1}{\cos^2 \theta} = \frac{1}{\left(\frac{3}{5}\right)^2} = \frac{25}{9}$

Simplifying the equation, we get:

$\tan^2 \theta = \frac{25}{9} - 1 = \frac{16}{9}$

Taking the square root of both sides, we get:

$\tan \theta = \pm \sqrt{\frac{16}{9}} = \pm \frac{4}{3}$

Since $\theta$ lies in Quadrant IV, the value of $\tan \theta$ is negative. Therefore, we can substitute the value of $\tan \theta$ into the equation:

$\tan 2\theta = \frac{2 \cdot \left(-\frac{4}{3}\right)}{1 - \left(-\frac{4}{3}\right)^2} = \frac{-\frac{8}{3}}{1 - \frac{16}{9}} = \frac{-\frac{8}{3}}{-\frac{7}{9}} = \frac{24}{7}$

Conclusion

In this article, we have solved three double-angle trigonometric expressions in Quadrant IV, given that $\cos \theta=\frac{3}{5}$ and $\theta$ lies in Quadrant IV. We have used the double-angle trigonometric identities to find the exact values of the double-angle trigonometric expressions. The solutions are:

• $\sin 2\theta = \frac{24}{25}$
• $\cos 2\theta = -\frac{7}{25}$
• $\tan 2\theta = \frac{24}{7}$

Introduction

In our previous article, we solved three double-angle trigonometric expressions in Quadrant IV, given that $\cos \theta=\frac{3}{5}$ and $\theta$ lies in Quadrant IV. In this article, we will provide a Q&A guide to help you understand the concepts and solutions better.

Q: What are double-angle trigonometric expressions?

A: Double-angle trigonometric expressions are used to find the values of trigonometric functions for double angles. These expressions are essential in solving various trigonometric problems, especially in calculus and engineering applications.

Q: What are the double-angle trigonometric identities?

A: The double-angle trigonometric identities are:

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

Q: How do I solve double-angle trigonometric expressions?

A: To solve double-angle trigonometric expressions, you need to use the double-angle trigonometric identities. You also need to find the values of sine, cosine, and tangent for the given angle.

Q: What is the significance of Quadrant IV in trigonometry?

A: Quadrant IV is one of the four quadrants in the Cartesian coordinate system. It is located in the upper-left region of the coordinate plane, where both the x-coordinate and y-coordinate are positive. In trigonometry, Quadrant IV is associated with angles that lie between $90^\circ$ and $180^\circ$.

Q: How do I find the values of sine, cosine, and tangent for a given angle?

A: To find the values of sine, cosine, and tangent for a given angle, you can use the Pythagorean identity:

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

You can also use the unit circle to find the values of sine, cosine, and tangent for a given angle.

Q: What are some common mistakes to avoid when solving double-angle trigonometric expressions?

A: Some common mistakes to avoid when solving double-angle trigonometric expressions include:

• Not using the correct double-angle trigonometric identity
• Not finding the values of sine, cosine, and tangent for the given angle
• Not simplifying the expression correctly
• Not checking the quadrant of the angle

Q: How can I practice solving double-angle trigonometric expressions?

A: You can practice solving double-angle trigonometric expressions by:

• Using online resources and practice problems
• Working with a tutor or teacher
• Practicing with real-world applications
• Joining a study group or online community

Conclusion

In this article, we have provided a Q&A guide to help you understand the concepts and solutions of double-angle trigonometric expressions. We have covered topics such as the significance of Quadrant IV, the double-angle trigonometric identities, and common mistakes to avoid. By practicing and understanding these concepts, you can become proficient in solving double-angle trigonometric expressions and apply them to real-world problems.

Additional Resources

• Khan Academy: Double-Angle Trigonometric Identities
• Mathway: Double-Angle Trigonometric Expressions
• Wolfram Alpha: Double-Angle Trigonometric Identities

Final Tips

• Practice regularly to improve your skills
• Use online resources and practice problems to reinforce your understanding
• Work with a tutor or teacher to get personalized feedback
• Join a study group or online community to connect with others who are learning the same material