Determining Trigonometric Functions Given Cotangent And Cosine Conditions

In this article, we will explore how to sketch the other five trigonometric functions when given information about one trigonometric function and the quadrant in which the angle lies. Specifically, we will focus on the example where $\cot \theta = -3$ and $\cos \theta > 0$. This problem requires a strong understanding of the relationships between trigonometric functions, their signs in different quadrants, and the fundamental trigonometric identities. By carefully analyzing the given information, we can systematically determine the values of $\sin \theta$, $\cos \theta$, $\tan \theta$, $\csc \theta$, and $\sec \theta$, and sketch their graphs accordingly.

Understanding the Given Information

The problem states that $\cot \theta = -3$ and $\cos \theta > 0$. Let's break down what each piece of information tells us:

• Cotangent ($\cot \theta = -3$): The cotangent function is the reciprocal of the tangent function, i.e., $\cot \theta = \frac{1}{\tan \theta}$. Since $\cot \theta$ is negative, this implies that $\tan \theta$ is also negative. Recall that $\tan \theta = \frac{\sin \theta}{\cos \theta}$, so a negative tangent means that either $\sin \theta$ is positive and $\cos \theta$ is negative, or $\sin \theta$ is negative and $\cos \theta$ is positive.
• Cosine ($\cos \theta > 0$): This tells us that the cosine of $\theta$ is positive. In the unit circle, cosine corresponds to the x-coordinate, so $\theta$ must lie in either the first or fourth quadrant.

Combining these two pieces of information, we can deduce the quadrant in which $\theta$ lies. Since $\cot \theta$ is negative and $\cos \theta$ is positive, $\theta$ must be in the fourth quadrant, where cosine is positive and sine (and therefore tangent and cotangent) is negative. This is a crucial first step in solving the problem, as the quadrant determines the signs of the trigonometric functions.

Determining the Values of Trigonometric Functions

Now that we know $\theta$ is in the fourth quadrant, we can find the values of the other trigonometric functions. We are given $\cot \theta = -3$, which can be written as $\cot \theta = \frac{-3}{1}$. Since $\cot \theta = \frac{\cos \theta}{\sin \theta}$, we can think of this as a ratio of the adjacent side to the opposite side in a right triangle, keeping in mind the signs in the fourth quadrant.

To find the values, we can use the Pythagorean identity, which relates sine and cosine: $\sin^2 \theta + \cos^2 \theta = 1$. We also know the identity $\cot^2 \theta + 1 = \csc^2 \theta$, which can help us find $\csc \theta$ directly.

1. Find $\csc \theta$: Using the identity $\cot^2 \theta + 1 = \csc^2 \theta$, we have $(-3)^2 + 1 = \csc^2 \theta$, which simplifies to $9 + 1 = \csc^2 \theta$, so $\csc^2 \theta = 10$. Therefore, $\csc \theta = \pm \sqrt{10}$. Since $\theta$ is in the fourth quadrant, where sine is negative, $\csc \theta$ (the reciprocal of sine) is also negative. Thus, $\csc \theta = -\sqrt{10}$.

2. Find $\sin \theta$: Since $\csc \theta = \frac{1}{\sin \theta}$, we have $\sin \theta = \frac{1}{\csc \theta} = \frac{1}{-\sqrt{10}} = -\frac{1}{\sqrt{10}}$. Rationalizing the denominator, we get $\sin \theta = -\frac{\sqrt{10}}{10}$.

3. Find $\tan \theta$: We know that $\cot \theta = -3$, and $\tan \theta$ is the reciprocal of $\cot \theta$. Therefore, $\tan \theta = \frac{1}{\cot \theta} = \frac{1}{-3} = -\frac{1}{3}$.

4. Find $\cos \theta$: We can use the identity $\tan \theta = \frac{\sin \theta}{\cos \theta}$ to find $\cos \theta$. We have $-\frac{1}{3} = \frac{-\frac{\sqrt{10}}{10}}{\cos \theta}$. Solving for $\cos \theta$, we get $\cos \theta = \frac{-\frac{\sqrt{10}}{10}}{-\frac{1}{3}} = \frac{\sqrt{10}}{10} \cdot 3 = \frac{3\sqrt{10}}{10}$.

5. Find $\sec \theta$: Since $\sec \theta = \frac{1}{\cos \theta}$, we have $\sec \theta = \frac{1}{\frac{3\sqrt{10}}{10}} = \frac{10}{3\sqrt{10}}$. Rationalizing the denominator, we get $\sec \theta = \frac{10\sqrt{10}}{3 \cdot 10} = \frac{\sqrt{10}}{3}$.

Now we have found all five trigonometric functions:

• $\sin \theta = -\frac{\sqrt{10}}{10}$
• $\cos \theta = \frac{3\sqrt{10}}{10}$
• $\tan \theta = -\frac{1}{3}$
• $\csc \theta = -\sqrt{10}$
• $\sec \theta = \frac{\sqrt{10}}{3}$

Sketching the Trigonometric Functions

Sketching the trigonometric functions involves understanding their graphs and key features. While a precise sketch requires plotting points, we can create a general sketch by knowing the quadrant, the signs of the functions, and their relative magnitudes.

1. Sine Function: Since $\sin \theta = -\frac{\sqrt{10}}{10}$, which is a negative value close to -0.316, we know the sine value is negative and relatively small in magnitude. In the fourth quadrant, the sine function starts at 0 at $2\pi$ (or 0 radians) and decreases to -1 at $\frac{3\pi}{2}$ radians. Our specific value is between these extremes. A sketch would show a sine wave in the fourth quadrant with a negative amplitude.

2. Cosine Function: $\cos \theta = \frac{3\sqrt{10}}{10}$ is a positive value, approximately 0.949, which is close to 1. In the fourth quadrant, the cosine function starts at 1 at $2\pi$ (or 0 radians) and decreases to 0 at $\frac{3\pi}{2}$ radians. Our value is close to the starting point, indicating the angle $\theta$ is closer to $2\pi$. The cosine sketch would show a wave starting near its maximum positive value.

3. Tangent Function: $\tan \theta = -\frac{1}{3}$ is a negative value, indicating the tangent is negative in this quadrant. In the fourth quadrant, tangent ranges from 0 at $2\pi$ to negative infinity as it approaches $\frac{3\pi}{2}$. Our value of $-\frac{1}{3}$ suggests the angle is not too close to $\frac{3\pi}{2}$. The tangent sketch will be a curve approaching the x-axis from negative values.

4. Cosecant Function: $\csc \theta = -\sqrt{10}$ is a negative value, approximately -3.16. Cosecant is the reciprocal of sine, so it has asymptotes where sine is zero and approaches infinity where sine approaches zero. The sketch would show a curve in the fourth quadrant, going towards negative infinity.

5. Secant Function: $\sec \theta = \frac{\sqrt{10}}{3}$ is a positive value, approximately 1.054. Secant is the reciprocal of cosine, so it has asymptotes where cosine is zero and approaches infinity where cosine approaches infinity. The sketch will be a curve in the fourth quadrant, close to its minimum positive value.

In summary, the graphs should reflect the signs and magnitudes of the trigonometric functions in the fourth quadrant, with attention to asymptotes for reciprocal functions (cosecant and secant). It's important to understand how these functions behave in different quadrants to accurately sketch them given specific information.

Conclusion

Determining the values and sketching trigonometric functions from given information, such as $\cot \theta = -3$ and $\cos \theta > 0$, involves a systematic approach. First, identify the quadrant in which the angle lies by considering the signs of the given functions. Then, use trigonometric identities and the definitions of the functions to find the values of the remaining functions. Finally, sketch the functions, paying attention to their signs, magnitudes, and asymptotes. This process reinforces the fundamental relationships between trigonometric functions and their graphical representations. Understanding how to approach these types of problems is crucial for mastering trigonometry and its applications in various fields of mathematics and science. This exercise not only solidifies the understanding of trigonometric identities but also enhances the ability to visualize and interpret trigonometric functions in different contexts. By working through such problems, one can develop a more intuitive grasp of the behavior of trigonometric functions, which is essential for more advanced topics in mathematics and physics. Ultimately, the ability to sketch trigonometric functions based on given information is a testament to a comprehensive understanding of trigonometric principles and their practical applications. The step-by-step process of identifying the quadrant, calculating the function values, and then sketching the curves provides a robust framework for solving similar problems and building a deeper appreciation for the elegance and utility of trigonometric functions.