Understanding Quadrant III

Oct 13, 2025 by thesweetdesigns.com 128 views

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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 given that $\cos \theta = -\frac{7}{25}$ and $\theta$ is in Quadrant III.

Understanding Quadrant III

Quadrant III is one of the four quadrants in the Cartesian coordinate system. It is located in the lower left corner of the coordinate plane and is characterized by negative values of both x and y coordinates. In this quadrant, the cosine function is negative, and the sine function is also negative.

Finding the Value of Sine

To find the value of sine, we can use the Pythagorean identity:

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

Since $\cos \theta = -\frac{7}{25}$ , we can substitute this value into the equation:

\sin^2 \theta + \left(-\frac{7}{25}\right)^2 = 1

Simplifying the equation, we get:

\sin^2 \theta + \frac{49}{625} = 1

Subtracting $\frac{49}{625}$ from both sides, we get:

\sin^2 \theta = \frac{576}{625}

Taking the square root of both sides, we get:

\sin \theta = \pm \sqrt{\frac{576}{625}}

Since $\theta$ is in Quadrant III, the sine function is negative. Therefore, we take the negative square root:

\sin \theta = -\sqrt{\frac{576}{625}}

Simplifying the expression, we get:

\sin \theta = -\frac{24}{25}

Finding the Value of Tangent

To find the value of tangent, we can use the definition of tangent:

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

Substituting the values of sine and cosine, we get:

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

Simplifying the expression, we get:

\tan \theta = \frac{24}{7}

Finding the Value of Cotangent

To find the value of cotangent, we can use the definition of cotangent:

\cot \theta = \frac{\cos \theta}{\sin \theta}

Substituting the values of sine and cosine, we get:

\cot \theta = \frac{-\frac{7}{25}}{-\frac{24}{25}}

Simplifying the expression, we get:

\cot \theta = \frac{7}{24}

Finding the Value of Secant

To find the value of secant, we can use the definition of secant:

\sec \theta = \frac{1}{\cos \theta}

Substituting the value of cosine, we get:

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

Simplifying the expression, we get:

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

Finding the Value of Cosecant

To find the value of cosecant, we can use the definition of cosecant:

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

Substituting the value of sine, we get:

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

Simplifying the expression, we get:

\csc \theta = -\frac{25}{24}

Conclusion

In this article, we have found the exact value of each of the remaining trigonometric functions given that $\cos \theta = -\frac{7}{25}$ and $\theta$ is in Quadrant III. We have used the Pythagorean identity to find the value of sine, and then used the definitions of tangent, cotangent, secant, and cosecant to find their values. The values of the trigonometric functions are:

$\sin \theta = -\frac{24}{25}$
$\tan \theta = \frac{24}{7}$
$\cot \theta = \frac{7}{24}$
$\sec \theta = -\frac{25}{7}$
$\csc \theta = -\frac{25}{24}$

These values can be used to solve problems involving trigonometry and can be applied in various fields, including physics, engineering, and navigation.

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Frequently Asked Questions

Q: What is the relationship between the trigonometric functions?

A: The trigonometric functions are related to each other through the Pythagorean identity:

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

This identity can be used to find the value of one trigonometric function if the value of another is known.

Q: How do I find the value of sine if I know the value of cosine?

A: To find the value of sine, you can use the Pythagorean identity:

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

Substitute the value of cosine into the equation and solve for sine.

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

A: To find the value of tangent, you can use the definition of tangent:

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

Substitute the values of sine and cosine into the equation and simplify.

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

A: To find the value of cotangent, you can use the definition of cotangent:

\cot \theta = \frac{\cos \theta}{\sin \theta}

Substitute the values of sine and cosine into the equation and simplify.

Q: How do I find the value of secant if I know the value of cosine?

A: To find the value of secant, you can use the definition of secant:

\sec \theta = \frac{1}{\cos \theta}

Substitute the value of cosine into the equation and simplify.

Q: How do I find the value of cosecant if I know the value of sine?

A: To find the value of cosecant, you can use the definition of cosecant:

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

Substitute the value of sine into the equation and simplify.

Q: What is the difference between the trigonometric functions?

A: The trigonometric functions are related to each other through the Pythagorean identity, but they have different definitions and properties. For example, the sine function is defined as the ratio of the opposite side to the hypotenuse in a right triangle, while the cosine function is defined as the ratio of the adjacent side to the hypotenuse.

Q: How do I use the trigonometric functions to solve problems?

A: The trigonometric functions can be used to solve problems involving right triangles, including finding the length of sides, the measure of angles, and the area of triangles. They can also be used to solve problems involving periodic functions, including finding the amplitude, period, and phase shift of a function.

Q: What are some common applications of the trigonometric functions?

A: The trigonometric functions have numerous applications in various fields, including physics, engineering, navigation, and computer science. They are used to model periodic phenomena, such as the motion of objects, the behavior of electrical circuits, and the patterns of sound waves.

Q: How do I choose the correct trigonometric function to use in a problem?

A: To choose the correct trigonometric function to use in a problem, you need to identify the type of problem and the information given. For example, if you are given the length of the hypotenuse and the measure of an angle, you would use the sine or cosine function to find the length of the opposite or adjacent side. If you are given the length of the opposite or adjacent side and the measure of an angle, you would use the tangent or cotangent function to find the length of the other side.

Q: What are some common mistakes to avoid when using the trigonometric functions?

A: Some common mistakes to avoid when using the trigonometric functions include:

Using the wrong trigonometric function for the problem
Not using the correct units for the problem
Not checking the domain and range of the function
Not simplifying the expression before solving the problem
Not checking the answer for reasonableness

Q: How do I check my answer for reasonableness?

A: To check your answer for reasonableness, you need to make sure that it is within the expected range and that it makes sense in the context of the problem. For example, if you are finding the length of a side in a right triangle, your answer should be a positive number. If you are finding the measure of an angle, your answer should be between 0 and 180 degrees.

Q: What are some resources for learning more about the trigonometric functions?

A: Some resources for learning more about the trigonometric functions include:

Textbooks and online resources, such as Khan Academy and MIT OpenCourseWare
Online tutorials and videos, such as 3Blue1Brown and Crash Course
Practice problems and worksheets, such as those found on Mathway and Wolfram Alpha
Online communities and forums, such as Reddit's r/learnmath and r/math