The Quadrant IV and the Sign of Trigonometric Functions

Oct 13, 2025 by miodeya.com 123 views

Understanding the Problem

When dealing with trigonometric functions, it's essential to understand the relationship between the sine and cosine of an angle. In this problem, we're given that the terminal side of $\theta$ is in quadrant IV and that $\sin \theta = -\frac{5}{13}$ . Our goal is to find the value of $\cos \theta$ .

The Quadrant IV and the Sign of Trigonometric Functions

Quadrant IV is the fourth quadrant in the unit circle, where the x-coordinate is positive and the y-coordinate is negative. In this quadrant, the sine function is negative, and the cosine function is positive. This is because the sine function is defined as the ratio of the y-coordinate to the hypotenuse, and the cosine function is defined as the ratio of the x-coordinate to the hypotenuse.

Using the Pythagorean Identity

We can use the Pythagorean identity to find the value of $\cos \theta$ . The Pythagorean identity states that $\sin^2 \theta + \cos^2 \theta = 1$ . We can rearrange this identity to solve for $\cos \theta$ : $\cos \theta = \pm \sqrt{1 - \sin^2 \theta}$ .

Finding the Value of $\cos \theta$

We're given that $\sin \theta = -\frac{5}{13}$ . We can substitute this value into the Pythagorean identity to find the value of $\cos \theta$ : $\cos \theta = \pm \sqrt{1 - \left(-\frac{5}{13}\right)^2}$ . Simplifying this expression, we get: $\cos \theta = \pm \sqrt{1 - \frac{25}{169}} = \pm \sqrt{\frac{144}{169}} = \pm \frac{12}{13}$ .

Determining the Sign of $\cos \theta$

Since the terminal side of $\theta$ is in quadrant IV, we know that the cosine function is positive in this quadrant. Therefore, the value of $\cos \theta$ must be positive.

Conclusion

Based on the Pythagorean identity and the fact that the terminal side of $\theta$ is in quadrant IV, we can conclude that the value of $\cos \theta$ is $\frac{12}{13}$ .

Final Answer

The final answer is $\boxed{\frac{12}{13}}$ .

Additional Information

It's worth noting that the other options are not correct. Option A, $-\frac{12}{13}$ , is the negative value of $\cos \theta$ , but we know that the cosine function is positive in quadrant IV. Option C, $\frac{5}{12}$ , is not a possible value for $\cos \theta$ given the information provided. Option D, $-\frac{5}{12}$ , is not a possible value for $\cos \theta$ given the information provided.

Common Mistakes

One common mistake when solving this problem is to forget to consider the sign of the cosine function in quadrant IV. It's essential to remember that the cosine function is positive in quadrant IV, which means that the value of $\cos \theta$ must be positive.

Real-World Applications

Understanding the relationship between the sine and cosine of an angle has many real-world applications. For example, in physics, the sine and cosine functions are used to describe the motion of objects in terms of their position, velocity, and acceleration. In engineering, the sine and cosine functions are used to describe the behavior of electrical circuits and mechanical systems.

Conclusion

In conclusion, the terminal side of $\theta$ is in quadrant IV, and $\sin \theta = -\frac{5}{13}$ . Using the Pythagorean identity and the fact that the cosine function is positive in quadrant IV, we can conclude that the value of $\cos \theta$ is $\frac{12}{13}$ .

Frequently Asked Questions

Q: What is the relationship between the sine and cosine of an angle?

A: The sine and cosine of an angle are related by the Pythagorean identity, which states that $\sin^2 \theta + \cos^2 \theta = 1$ . This identity can be used to find the value of one trigonometric function given the value of the other.

Q: How do you determine the sign of the cosine function in a given quadrant?

A: The sign of the cosine function is determined by the x-coordinate of the point on the unit circle. In quadrant I, the x-coordinate is positive, so the cosine function is positive. In quadrant II, the x-coordinate is negative, so the cosine function is negative. In quadrant III, the x-coordinate is negative, so the cosine function is negative. In quadrant IV, the x-coordinate is positive, so the cosine function is positive.

Q: What is the value of $\cos \theta$ when $\sin \theta = -\frac{5}{13}$ and the terminal side of $\theta$ is in quadrant IV?

A: Using the Pythagorean identity, we can find the value of $\cos \theta$ as follows: $\cos \theta = \pm \sqrt{1 - \sin^2 \theta} = \pm \sqrt{1 - \left(-\frac{5}{13}\right)^2} = \pm \sqrt{1 - \frac{25}{169}} = \pm \sqrt{\frac{144}{169}} = \pm \frac{12}{13}$ . Since the terminal side of $\theta$ is in quadrant IV, we know that the cosine function is positive, so the value of $\cos \theta$ is $\frac{12}{13}$ .

Q: What are some common mistakes to avoid when solving this problem?

A: One common mistake is to forget to consider the sign of the cosine function in quadrant IV. It's essential to remember that the cosine function is positive in quadrant IV, which means that the value of $\cos \theta$ must be positive. Another common mistake is to substitute the value of $\sin \theta$ into the Pythagorean identity without considering the sign of the cosine function.

Q: What are some real-world applications of the sine and cosine functions?

A: The sine and cosine functions have many real-world applications. In physics, the sine and cosine functions are used to describe the motion of objects in terms of their position, velocity, and acceleration. In engineering, the sine and cosine functions are used to describe the behavior of electrical circuits and mechanical systems.

Q: How can I use the Pythagorean identity to find the value of one trigonometric function given the value of the other?

A: To use the Pythagorean identity to find the value of one trigonometric function given the value of the other, you can substitute the value of the known function into the identity and solve for the unknown function. For example, if you know the value of $\sin \theta$ and want to find the value of $\cos \theta$ , you can substitute the value of $\sin \theta$ into the Pythagorean identity and solve for $\cos \theta$ .

Q: What is the significance of the unit circle in trigonometry?

A: The unit circle is a fundamental concept in trigonometry, as it provides a visual representation of the relationships between the sine and cosine functions. The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. The points on the unit circle correspond to the values of the sine and cosine functions for different angles.

Q: How can I use the unit circle to find the value of the sine and cosine functions for different angles?

A: To use the unit circle to find the value of the sine and cosine functions for different angles, you can locate the point on the unit circle corresponding to the given angle and read off the values of the sine and cosine functions. Alternatively, you can use the Pythagorean identity to find the value of one trigonometric function given the value of the other.

Q: What are some common trigonometric identities that I should know?

A: Some common trigonometric identities that you should know include the Pythagorean identity, the sum and difference identities, and the double-angle and half-angle identities. These identities can be used to simplify trigonometric expressions and solve trigonometric equations.

Q: How can I use trigonometric identities to simplify trigonometric expressions?

A: To use trigonometric identities to simplify trigonometric expressions, you can substitute the values of the trigonometric functions into the identity and simplify the resulting expression. For example, if you have the expression $\sin^2 \theta + \cos^2 \theta$ , you can substitute the values of $\sin \theta$ and $\cos \theta$ into the Pythagorean identity and simplify the resulting expression to get 1.

Q: What are some real-world applications of trigonometry?

A: Trigonometry has many real-world applications, including physics, engineering, navigation, and computer graphics. In physics, trigonometry is used to describe the motion of objects in terms of their position, velocity, and acceleration. In engineering, trigonometry is used to describe the behavior of electrical circuits and mechanical systems. In navigation, trigonometry is used to determine the position and orientation of objects in space. In computer graphics, trigonometry is used to create 3D models and animations.