Recalling the Pythagorean Identity

Oct 13, 2025 by windailys.com 97 views

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject in mathematics and has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will focus on finding the exact value of each of the remaining trigonometric functions of $\theta$ , given that $\cos \theta = -\frac{7}{25}$ and $\sin \theta = \frac{24}{25}$ , with $\theta$ being in Quadrant II.

Understanding Quadrant II

Quadrant II is one of the four quadrants in the Cartesian coordinate system. It is located in the upper-left region of the coordinate plane and is characterized by positive $x$ -coordinates and negative $y$ -coordinates. In the context of trigonometry, Quadrant II is associated with angles that lie between $90^\circ$ and $180^\circ$ . Since $\theta$ is in Quadrant II, we know that $\sin \theta$ is positive and $\cos \theta$ is negative.

Recalling the Pythagorean Identity

The Pythagorean identity is a fundamental relationship between the sine, cosine, and tangent functions. It states that:

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

This identity can be used to find the value of one trigonometric function if the values of the other two functions are known.

Finding the Value of $\tan \theta$

We are given that $\cos \theta = -\frac{7}{25}$ and $\sin \theta = \frac{24}{25}$ . We can use the Pythagorean identity to find the value of $\tan \theta$ .

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

Therefore, the exact value of $\tan \theta$ is $-\frac{24}{7}$ .

Finding the Value of $\csc \theta$

The cosecant function is the reciprocal of the sine function. Therefore, we can find the value of $\csc \theta$ by taking the reciprocal of $\sin \theta$ .

\csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{24}{25}} = \frac{25}{24}

Therefore, the exact value of $\csc \theta$ is $\frac{25}{24}$ .

Finding the Value of $\sec \theta$

The secant function is the reciprocal of the cosine function. Therefore, we can find the value of $\sec \theta$ by taking the reciprocal of $\cos \theta$ .

\sec \theta = \frac{1}{\cos \theta} = \frac{1}{-\frac{7}{25}} = -\frac{25}{7}

Therefore, the exact value of $\sec \theta$ is $-\frac{25}{7}$ .

Finding the Value of $\cot \theta$

The cotangent function is the reciprocal of the tangent function. Therefore, we can find the value of $\cot \theta$ by taking the reciprocal of $\tan \theta$ .

\cot \theta = \frac{1}{\tan \theta} = \frac{1}{-\frac{24}{7}} = -\frac{7}{24}

Therefore, the exact value of $\cot \theta$ is $-\frac{7}{24}$ .

Conclusion

In this article, we have found the exact value of each of the remaining trigonometric functions of $\theta$ , given that $\cos \theta = -\frac{7}{25}$ and $\sin \theta = \frac{24}{25}$ , with $\theta$ being in Quadrant II. We have used the Pythagorean identity and the reciprocal relationships between the trigonometric functions to find the values of $\tan \theta$ , $\csc \theta$ , $\sec \theta$ , and $\cot \theta$ . These values are essential in various mathematical and scientific applications, and understanding how to find them is crucial for success in these fields.

References

[1] "Trigonometry" by Michael Corral, 2019.
[2] "Calculus" by Michael Spivak, 2008.
[3] "Mathematics for Engineers and Scientists" by Donald R. Hill, 2017.

Glossary

Quadrant II: The upper-left region of the Cartesian coordinate system, characterized by positive $x$ -coordinates and negative $y$ -coordinates.
Pythagorean identity: A fundamental relationship between the sine, cosine, and tangent functions, stating that $\sin^2 \theta + \cos^2 \theta = 1$ .
Reciprocal relationships: The relationships between the trigonometric functions, where the reciprocal of one function is equal to the other function.
Trigonometry Q&A: Finding the Exact Value of Trigonometric Functions ====================================================================

Introduction

In our previous article, we discussed how to find the exact value of each of the remaining trigonometric functions of $\theta$ , given that $\cos \theta = -\frac{7}{25}$ and $\sin \theta = \frac{24}{25}$ , with $\theta$ being in Quadrant II. In this article, we will answer some frequently asked questions related to trigonometry and provide additional examples to help you understand the concepts better.

Q: What is the difference between the sine and cosine functions?

A: The sine and cosine functions are two of the most fundamental trigonometric functions. The sine function represents the ratio of the length of the side opposite the angle to the length of the hypotenuse, while the cosine function represents the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

Q: How do I remember the values of the sine and cosine functions for common angles?

A: There are several ways to remember the values of the sine and cosine functions for common angles. One way is to use the unit circle, which is a circle with a radius of 1 centered at the origin of the coordinate plane. The sine and cosine functions can be represented as the $y$ -coordinate and $x$ -coordinate of a point on the unit circle, respectively.

Q: What is the Pythagorean identity, and how is it used in trigonometry?

A: The Pythagorean identity is a fundamental relationship between the sine, cosine, and tangent functions, stating that $\sin^2 \theta + \cos^2 \theta = 1$ . This identity can be used to find the value of one trigonometric function if the values of the other two functions are known.

Q: How do I find the value of the tangent function if I know the values of the sine and cosine functions?

A: To find the value of the tangent function, you can use the formula $\tan \theta = \frac{\sin \theta}{\cos \theta}$ . This formula represents the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Q: What is the difference between the cosecant and secant functions?

A: The cosecant function is the reciprocal of the sine function, while the secant function is the reciprocal of the cosine function. This means that the cosecant function represents the ratio of the length of the hypotenuse to the length of the side opposite the angle, while the secant function represents the ratio of the length of the hypotenuse to the length of the side adjacent to the angle.

Q: How do I find the value of the cotangent function if I know the values of the sine and cosine functions?

A: To find the value of the cotangent function, you can use the formula $\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}$ . This formula represents the ratio of the length of the side adjacent to the angle to the length of the side opposite the angle.

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

A: There are several common trigonometric identities that you should know, including:

The Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$
The sum and difference formulas: $\sin (A + B) = \sin A \cos B + \cos A \sin B$ and $\sin (A - B) = \sin A \cos B - \cos A \sin B$
The double-angle formulas: $\sin 2\theta = 2\sin \theta \cos \theta$ and $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$

Conclusion

In this article, we have answered some frequently asked questions related to trigonometry and provided additional examples to help you understand the concepts better. We hope that this article has been helpful in clarifying any confusion you may have had about trigonometry.

References

[1] "Trigonometry" by Michael Corral, 2019.
[2] "Calculus" by Michael Spivak, 2008.
[3] "Mathematics for Engineers and Scientists" by Donald R. Hill, 2017.

Glossary

Unit circle: A circle with a radius of 1 centered at the origin of the coordinate plane.
Pythagorean identity: A fundamental relationship between the sine, cosine, and tangent functions, stating that $\sin^2 \theta + \cos^2 \theta = 1$ .
Reciprocal relationships: The relationships between the trigonometric functions, where the reciprocal of one function is equal to the other function.