Finding Cos Θ, Sec Θ, And Cot Θ For Point (-5, 6)

Trigonometric functions play a fundamental role in mathematics, physics, and engineering. Understanding how to determine the exact values of these functions is crucial for solving a wide range of problems. One common scenario involves finding the trigonometric values when given a point on the terminal side of an angle in standard position. In this article, we will explore how to find the exact values of cosine, secant, and cotangent when given the point $(-5, 6)$ on the terminal side of an angle $\theta$.

Understanding the Basics

Before we dive into the specifics, let's review some key concepts. An angle in standard position has its vertex at the origin and its initial side along the positive x-axis. The terminal side is the ray that rotates from the initial side to form the angle. A point on the terminal side, along with the origin, forms a right triangle. The sides of this triangle are related to the x-coordinate, y-coordinate, and the distance from the origin to the point.

• Coordinates and the Radius: If $(x, y)$ is a point on the terminal side of an angle $\theta$ in standard position, we can define $r$ as the distance from the origin to the point $(x, y)$. This distance $r$ is also known as the radius and can be calculated using the Pythagorean theorem: $r = \sqrt{x^2 + y^2}$.
• Trigonometric Ratios: The trigonometric functions are defined as ratios of the sides of the right triangle formed by the point $(x, y)$, the origin, and the x-axis. The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan). Their reciprocals are cosecant (csc), secant (sec), and cotangent (cot), respectively.
  • $\sin \theta = \frac{y}{r}$
  • $\cos \theta = \frac{x}{r}$
  • $\tan \theta = \frac{y}{x}$
  • $\csc \theta = \frac{r}{y}$
  • $\sec \theta = \frac{r}{x}$
  • $\cot \theta = \frac{x}{y}$

These ratios allow us to relate the angle $\theta$ to the coordinates of any point on its terminal side. By understanding these relationships, we can find the exact values of the trigonometric functions.

Step-by-Step Solution for the Given Problem

Now, let's apply these concepts to our specific problem. We are given the point $(-5, 6)$ on the terminal side of an angle $\theta$, and we need to find the exact values of $\cos \theta$, $\sec \theta$, and $\cot \theta$.

Step 1: Calculate the Radius (r)

First, we need to find the distance $r$ from the origin to the point $(-5, 6)$. We use the formula $r = \sqrt{x^2 + y^2}$:

$r = \sqrt{(-5)^2 + (6)^2} = \sqrt{25 + 36} = \sqrt{61}$

So, the radius $r$ is $\sqrt{61}$. This value is essential for calculating the trigonometric ratios.

Step 2: Determine Cosine (cos θ)

The cosine of $\theta$ is defined as $\cos \theta = \frac{x}{r}$. We know that $x = -5$ and $r = \sqrt{61}$, so:

$\cos \theta = \frac{-5}{\sqrt{61}}$

To rationalize the denominator, we multiply both the numerator and the denominator by $\sqrt{61}$:

$\cos \theta = \frac{-5}{\sqrt{61}} \cdot \frac{\sqrt{61}}{\sqrt{61}} = \frac{-5\sqrt{61}}{61}$

Thus, the exact value of $\cos \theta$ is $\frac{-5\sqrt{61}}{61}$.

Step 3: Find Secant (sec θ)

The secant is the reciprocal of the cosine function. Therefore, $\sec \theta = \frac{1}{\cos \theta} = \frac{r}{x}$. We already know that $r = \sqrt{61}$ and $x = -5$, so:

$\sec \theta = \frac{\sqrt{61}}{-5}$

This is the exact value of $\sec \theta$. It's the reciprocal of the cosine value we found earlier.

Step 4: Calculate Cotangent (cot θ)

The cotangent of $\theta$ is defined as $\cot \theta = \frac{x}{y}$. We have $x = -5$ and $y = 6$, so:

$\cot \theta = \frac{-5}{6}$

This gives us the exact value of $\cot \theta$. It represents the ratio of the x-coordinate to the y-coordinate of the point on the terminal side.

Summarizing the Results

We have successfully found the exact values of $\cos \theta$, $\sec \theta$, and $\cot \theta$ given the point $(-5, 6)$ on the terminal side of the angle $\theta$:

• $\cos \theta = \frac{-5\sqrt{61}}{61}$
• $\sec \theta = \frac{\sqrt{61}}{-5}$
• $\cot \theta = \frac{-5}{6}$

These values provide a comprehensive understanding of the trigonometric functions associated with the angle $\theta$. Understanding these functions is critical for various applications in mathematics and related fields.

Additional Practice and Applications

Practice Problems

To reinforce your understanding, try solving similar problems. For instance, find the exact values of $\sin \theta$, $\csc \theta$, and $\tan \theta$ for the same point $(-5, 6)$. You can also try other points in different quadrants to see how the signs of the trigonometric functions change.

Real-World Applications

Trigonometric functions are essential in many real-world applications, including:

• Physics: Analyzing projectile motion, oscillations, and waves.
• Engineering: Designing structures, analyzing circuits, and signal processing.
• Navigation: Calculating distances and directions.
• Computer Graphics: Creating realistic images and animations.

By mastering the fundamental concepts of trigonometry, you can unlock a wide range of problem-solving capabilities.

Conclusion

In this article, we have demonstrated how to find the exact values of $\cos \theta$, $\sec \theta$, and $\cot \theta$ when given a point on the terminal side of an angle. By understanding the definitions of trigonometric functions and applying the Pythagorean theorem, we can solve these problems systematically. The ability to find these values is a cornerstone of trigonometry and is crucial for more advanced topics in mathematics and its applications. Keep practicing, and you'll become proficient in working with trigonometric functions.

Understanding Trigonometric Functions

When diving into the world of trigonometry, it's essential to grasp the fundamental concept: the trigonometric functions. These functions, including sine, cosine, tangent, and their reciprocals (cosecant, secant, and cotangent), are pivotal in relating angles to the sides of a right triangle. One common task is determining the exact values of these trigonometric functions when we are given a specific point on the terminal side of an angle. So, let’s explore how to find $\cos \theta$, $\sec \theta$, and $\cot \theta$ when the terminal side of $\theta$ passes through the point $(-5, 6)$. This problem illustrates a crucial application of trigonometric definitions and the Pythagorean theorem.

To begin, recall the definitions of the primary trigonometric functions in terms of a point $(x, y)$ on the terminal side of an angle $\theta$ in standard position. The distance $r$ from the origin to the point $(x, y)$ is calculated as $r = \sqrt{x^2 + y^2}$. This value $r$ acts as the hypotenuse of the right triangle formed by the x-coordinate, y-coordinate, and the origin. The trigonometric functions are then defined as ratios of these sides. Specifically:

• $\sin \theta = \frac{y}{r}$
• $\cos \theta = \frac{x}{r}$
• $\tan \theta = \frac{y}{x}$

The reciprocal trigonometric functions are:

• $\csc \theta = \frac{r}{y}$
• $\sec \theta = \frac{r}{x}$
• $\cot \theta = \frac{x}{y}$

Understanding these ratios is the first step in solving problems involving trigonometric functions. In our case, we are given the point $(-5, 6)$, so $x = -5$ and $y = 6$. The goal is to find $\cos \theta$, $\sec \theta$, and $\cot \theta$ using these definitions. By applying the Pythagorean theorem, we can find the value of $r$, which is the distance from the origin to the point $(-5, 6)$. This distance is crucial for determining the trigonometric ratios. The sign of each trigonometric function depends on the quadrant in which the terminal side of the angle lies. This is a critical aspect to consider when calculating these values. Understanding the relationships between the x-coordinate, y-coordinate, and $r$ is fundamental to mastering trigonometry.

Step-by-Step Solution for cos θ, sec θ, and cot θ

Let’s dive into the step-by-step solution to find the exact values of $\cos \theta$, $\sec \theta$, and $\cot \theta$ given the point $(-5, 6)$ on the terminal side of $\theta$. This process will involve calculating $r$, and then applying the definitions of the trigonometric functions. Each step is crucial to ensure accuracy in the final results.

Step 1: Calculate the Radius (r)

The first step is to determine the distance $r$ from the origin to the point $(-5, 6)$. As we discussed, this is done using the Pythagorean theorem. The formula is $r = \sqrt{x^2 + y^2}$. Substituting the given values, we have:

$r = \sqrt{(-5)^2 + (6)^2} = \sqrt{25 + 36} = \sqrt{61}$

So, $r = \sqrt{61}$. This value represents the hypotenuse of the right triangle formed by the point $(-5, 6)$, the origin, and the x-axis. The value of $r$ is essential for calculating the trigonometric ratios because it appears in the denominators of $\sin \theta$, $\cos \theta$, $\csc \theta$, and $\sec \theta$. It’s important to note that $r$ is always positive since it represents a distance. With $r$ calculated, we can proceed to find the specific trigonometric values we are looking for.

Step 2: Determine Cosine (cos θ)

The cosine of $\theta$ is defined as the ratio of the x-coordinate to $r$, that is, $\cos \theta = \frac{x}{r}$. We have $x = -5$ and $r = \sqrt{61}$, so we can write:

$\cos \theta = \frac{-5}{\sqrt{61}}$

While this is technically a correct answer, it’s common practice to rationalize the denominator. To do this, we multiply both the numerator and the denominator by $\sqrt{61}$:

$\cos \theta = \frac{-5}{\sqrt{61}} \cdot \frac{\sqrt{61}}{\sqrt{61}} = \frac{-5\sqrt{61}}{61}$

Thus, the exact value of $\cos \theta$ is $\frac{-5\sqrt{61}}{61}$. This value tells us the ratio of the x-coordinate to the distance from the origin, and the negative sign indicates that the angle’s terminal side is in the second or third quadrant. Understanding how to rationalize the denominator is a crucial skill in trigonometry and ensures that the answer is in its simplest form. Now that we have $\cos \theta$, we can easily find $\sec \theta$, as it is the reciprocal of cosine.

Step 3: Find Secant (sec θ)

The secant function, $\sec \theta$, is the reciprocal of the cosine function. Therefore, $\sec \theta = \frac{1}{\cos \theta}$. Since we have already found that $\cos \theta = \frac{-5\sqrt{61}}{61}$, we can find $\sec \theta$ by taking the reciprocal:

$\sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{-5\sqrt{61}}{61}} = \frac{61}{-5\sqrt{61}}$

However, to rationalize the denominator, it's easier to take the reciprocal of the unrationalized form of $\cos \theta$, which was $\frac{-5}{\sqrt{61}}$. So:

$\sec \theta = \frac{1}{\frac{-5}{\sqrt{61}}} = \frac{\sqrt{61}}{-5}$

This gives us the exact value of $\sec \theta$. It is the reciprocal of $\cos \theta$, and like $\cos \theta$, it has a negative sign due to the quadrant in which the terminal side lies. The secant function represents the ratio of the distance from the origin to the x-coordinate. With $\sec \theta$ determined, we move on to find $\cot \theta$, which is the reciprocal of the tangent function.

Step 4: Calculate Cotangent (cot θ)

The cotangent of $\theta$ is defined as the ratio of the x-coordinate to the y-coordinate, $\cot \theta = \frac{x}{y}$. We are given $x = -5$ and $y = 6$, so:

$\cot \theta = \frac{-5}{6}$

This gives us the exact value of $\cot \theta$. There is no need to rationalize the denominator in this case since the denominator is already a rational number. The negative sign indicates that the angle’s terminal side is in either the second or fourth quadrant. The cotangent function is the reciprocal of the tangent function and represents the ratio of the adjacent side to the opposite side in the right triangle formed by the point, the origin, and the x-axis. Now, we have found the values of $\cos \theta$, $\sec \theta$, and $\cot \theta$ using the given point $(-5, 6)$.

Final Results and Synthesis

We have successfully calculated the exact values of $\cos \theta$, $\sec \theta$, and $\cot \theta$ given that the terminal side of $\theta$ passes through the point $(-5, 6)$. Let’s summarize our findings:

• $\cos \theta = \frac{-5\sqrt{61}}{61}$
• $\sec \theta = \frac{\sqrt{61}}{-5}$
• $\cot \theta = \frac{-5}{6}$

These values provide a complete picture of these trigonometric functions for the given angle. Understanding the relationships between trigonometric functions and the coordinates of points on the terminal side is essential for more advanced topics in trigonometry and calculus. Furthermore, recognizing the signs of these functions based on the quadrant is a critical skill. The point $(-5, 6)$ lies in the second quadrant, where cosine, secant, and cotangent are negative, which aligns with our results. By following a step-by-step approach and utilizing the definitions of trigonometric functions, you can accurately find these values for any given point.

In conclusion, this article has provided a detailed explanation of how to find the exact values of $\cos \theta$, $\sec \theta$, and $\cot \theta$ when given a point on the terminal side of an angle. By applying the Pythagorean theorem and the definitions of trigonometric functions, we were able to systematically solve the problem. Mastering these fundamental concepts is crucial for success in trigonometry and related fields such as calculus, physics, and engineering. Understanding the relationships between angles, coordinates, and trigonometric functions allows for a deeper comprehension of mathematical principles and their real-world applications. Keep practicing and reinforcing your knowledge to excel in this fascinating area of mathematics. Remember, each step in trigonometry builds upon the previous one, and a solid foundation will lead to greater proficiency and confidence.