Trigonometric Functions In Quadrant III Solving For Cosine

In the realm of trigonometry, understanding the relationship between trigonometric functions and the quadrants of the coordinate plane is crucial. When we're given a specific trigonometric value and the quadrant in which the angle's terminal point lies, we can determine the values of other trigonometric functions. Let's delve into a problem where $\cot \theta = \frac{3}{4}$ and the terminal point of $ heta$ is in Quadrant III. Our goal is to determine the values of other trigonometric functions and understand why the correct answer is $\cos \theta = -\frac{3}{5}$.

Understanding the Given Information

To effectively solve this problem, we must first break down the given information. We are told that the cotangent of angle $ heta$ is $\frac{3}{4}$. Recall that the cotangent function is defined as the ratio of the adjacent side to the opposite side in a right triangle, or equivalently, as $\frac{x}{y}$ in the coordinate plane. Additionally, we know that the terminal point of $\theta$ lies in Quadrant III. This is a critical piece of information because it tells us about the signs of the x and y coordinates. In Quadrant III, both x and y coordinates are negative.

Given that $\cot \theta = \frac{3}{4} = \frac{x}{y}$, we can initially think of x as 3 and y as 4. However, since we are in Quadrant III, both x and y must be negative. Therefore, we can represent x as -3 and y as -4. This understanding is crucial for correctly determining the signs of other trigonometric functions.

Visualizing the Triangle

To further solidify our understanding, let's visualize a right triangle in the coordinate plane. The angle $\theta$ is formed by the terminal side of the angle and the negative x-axis. The point (-3, -4) lies on the terminal side of the angle. We can form a right triangle by drawing a perpendicular line from this point to the x-axis. The legs of the triangle have lengths |-3| = 3 and |-4| = 4. We can find the length of the hypotenuse, r, using the Pythagorean theorem:

$$r = \sqrt{x^2 + y^2} = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$

Now we have all three sides of the right triangle: the adjacent side (x = -3), the opposite side (y = -4), and the hypotenuse (r = 5). With this information, we can determine the values of all six trigonometric functions.

Determining Other Trigonometric Functions

Now that we have the values of x, y, and r, we can calculate the other trigonometric functions. Let's start with sine, cosine, and cosecant.

• Sine ($\sin \theta$): Sine is defined as the ratio of the opposite side to the hypotenuse, or $\frac{y}{r}$. In this case, $\sin \theta = \frac{-4}{5} = -\frac{4}{5}$.
• Cosine ($\cos \theta$): Cosine is defined as the ratio of the adjacent side to the hypotenuse, or $\frac{x}{r}$. In this case, $\cos \theta = \frac{-3}{5} = -\frac{3}{5}$.
• Cosecant ($\csc \theta$): Cosecant is the reciprocal of sine, so $\csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{4}{5}} = -\frac{5}{4}$.

We already know the cotangent, which is given as $\cot \theta = \frac{3}{4}$. Let's also calculate the tangent and secant functions.

• Tangent ($\tan \theta$): Tangent is the reciprocal of cotangent, so $\tan \theta = \frac{1}{\cot \theta} = \frac{1}{\frac{3}{4}} = \frac{4}{3}$. Alternatively, tangent can be calculated as $\frac{y}{x} = \frac{-4}{-3} = \frac{4}{3}$.
• Secant ($\sec \theta$): Secant is the reciprocal of cosine, so $\sec \theta = \frac{1}{\cos \theta} = \frac{1}{-\frac{3}{5}} = -\frac{5}{3}$.

Analyzing the Options

Now let's compare our calculated values with the options given:

• **A. $\csc \theta = -\frac5}{3}$** We calculated $\csc \theta = -\frac{5{4}$, so this option is incorrect.
• **B. $\sin \theta = \frac3}{5}$** We calculated $\sin \theta = -\frac{4{5}$, so this option is incorrect.
• **C. $\cos \theta = -\frac3}{5}$** We calculated $\cos \theta = -\frac{3{5}$, so this option is correct.

Therefore, the correct answer is C. $\cos \theta = -\frac{3}{5}$. This process illustrates how understanding the definitions of trigonometric functions and the implications of the quadrant in which the angle lies allows us to solve such problems effectively.

Importance of Quadrant Information

The quadrant information is not just an extra detail; it is crucial for determining the signs of the trigonometric functions. Each quadrant has a specific pattern of positive and negative signs for sine, cosine, and tangent, which subsequently affects the signs of their reciprocals (cosecant, secant, and cotangent). Here's a quick recap:

• Quadrant I (0° - 90°): All trigonometric functions are positive.
• Quadrant II (90° - 180°): Sine and cosecant are positive; others are negative.
• Quadrant III (180° - 270°): Tangent and cotangent are positive; others are negative.
• Quadrant IV (270° - 360°): Cosine and secant are positive; others are negative.

In our problem, knowing that the terminal point is in Quadrant III immediately tells us that sine and cosine must be negative, while tangent and cotangent must be positive. This helps us avoid mistakes when calculating the trigonometric values.

Common Mistakes and How to Avoid Them

One common mistake in these types of problems is ignoring the quadrant information. If we had only considered the cotangent value of $\frac{3}{4}$ without considering Quadrant III, we might have incorrectly assumed positive values for both x and y, leading to incorrect values for sine and cosine. To avoid this, always start by identifying the quadrant and noting which trigonometric functions are positive and negative in that quadrant.

Another mistake is incorrectly applying the Pythagorean theorem or miscalculating the value of the hypotenuse. Double-checking the calculations and ensuring the correct signs are used will prevent these errors.

Conclusion: Mastering Trigonometric Functions and Quadrants

In conclusion, determining trigonometric function values given one value and the quadrant requires a solid understanding of the definitions of the trigonometric functions, the Pythagorean theorem, and the sign conventions in each quadrant. By carefully analyzing the given information, visualizing the triangle in the coordinate plane, and systematically calculating the values, we can confidently solve these problems.

In the given problem, with $&cot \theta = \frac{3}{4}$ and the terminal point in Quadrant III, we correctly determined that $&cos \theta = -\frac{3}{5}$. This exercise underscores the importance of paying attention to all details provided in the problem and applying the fundamental principles of trigonometry accurately. Understanding these principles not only helps in solving problems but also builds a strong foundation for more advanced topics in mathematics and physics.

By focusing on understanding rather than memorization, you can master trigonometric functions and their applications in various contexts. Remember to practice regularly, visualize the problems, and always double-check your work to ensure accuracy. This comprehensive approach will pave the way for success in trigonometry and beyond.