Finding Sin Θ Given Cot Θ = -2/5 And Θ In Quadrant IV

In trigonometry, determining the values of trigonometric functions within specific quadrants is a fundamental skill. This article delves into a problem where we are given the cotangent of an angle θ and the quadrant in which θ lies, and our objective is to find the sine of θ. Specifically, we are given that cot θ = -2/5 and that θ is in quadrant IV. We will walk through the steps to find sin θ, emphasizing the trigonometric identities and quadrant rules that govern such problems. Understanding these principles is crucial not only for solving mathematical problems but also for grasping the broader applications of trigonometry in fields like physics and engineering.

Before diving into the solution, let's clarify the given information. We know that cot θ, which is the cotangent of the angle θ, is equal to -2/5. The cotangent function is defined as the ratio of the adjacent side to the opposite side in a right-angled triangle, or equivalently, as the ratio of cosine to sine (cot θ = cos θ / sin θ). The negative sign indicates that either the cosine or the sine (but not both) must be negative, as the cotangent is negative in quadrants II and IV. We are also told that θ is in quadrant IV. This piece of information is crucial because it tells us the signs of the trigonometric functions in this quadrant. In quadrant IV, cosine is positive, while sine is negative. This is because in the Cartesian plane, quadrant IV is where the x-coordinates are positive and the y-coordinates are negative, corresponding to cosine and sine, respectively.

To solve this problem, we will primarily use two key trigonometric concepts: trigonometric identities and quadrant rules.

• Trigonometric Identities are equations involving trigonometric functions that are true for all values of the variables for which the functions are defined. The most important identity for this problem is the Pythagorean identity: sin² θ + cos² θ = 1. This identity relates sine and cosine, allowing us to find one if we know the other. We will also use the identity relating cotangent to sine and cosine: cot θ = cos θ / sin θ.
• Quadrant Rules dictate the signs of trigonometric functions in each quadrant of the Cartesian plane. As mentioned earlier, in quadrant IV, cosine is positive, and sine is negative. Tangent and cotangent, being ratios of sine and cosine, are also negative in quadrant IV. These rules are derived from the definitions of trigonometric functions in terms of the unit circle, where the x-coordinate corresponds to cosine, and the y-coordinate corresponds to sine.

1. Relate Cotangent to Sine and Cosine: We know that cot θ = cos θ / sin θ = -2/5. This equation provides a relationship between cos θ and sin θ, which we will use in conjunction with the Pythagorean identity.

2. Use the Pythagorean Identity: The Pythagorean identity states that sin² θ + cos² θ = 1. To use this identity, we need to express cos θ in terms of sin θ (or vice versa) using the given cotangent value. From cot θ = -2/5, we can write cos θ = (-2/5) sin θ. This expression allows us to substitute cos θ in the Pythagorean identity.

3. Substitute and Solve for Sine: Substitute cos θ = (-2/5) sin θ into the Pythagorean identity: sin² θ + ((-2/5) sin θ)² = 1 Simplify the equation: sin² θ + (4/25) sin² θ = 1 Combine the terms: (29/25) sin² θ = 1 Multiply both sides by 25/29: sin² θ = 25/29 Take the square root of both sides: sin θ = ±√(25/29) sin θ = ±5/√29 To rationalize the denominator, multiply the numerator and denominator by √29: sin θ = ±(5√29)/29

4. Determine the Sign: Since θ is in quadrant IV, sine is negative. Therefore, we choose the negative value: sin θ = -(5√29)/29

Therefore, given that cot θ = -2/5 and θ is in quadrant IV, sin θ = -(5√29)/29. This result is obtained by using the relationship between cotangent, sine, and cosine, along with the Pythagorean identity and the quadrant rules that dictate the signs of trigonometric functions in different quadrants.

In this article, we successfully found the value of sin θ given cot θ and the quadrant in which θ lies. The key to solving such problems is a solid understanding of trigonometric identities, quadrant rules, and the relationships between trigonometric functions. By relating cotangent to sine and cosine, using the Pythagorean identity, and considering the sign of sine in quadrant IV, we arrived at the final answer. This process underscores the interconnectedness of trigonometric concepts and their application in problem-solving. Understanding these principles is essential for further studies in mathematics, physics, and engineering, where trigonometric functions play a crucial role.