Finding Sin(θ) With Cosine And Quadrant Information

Hey guys! Let's dive into a classic trigonometry problem. We're given that cos(θ) = 13/85, and we know that the angle θ's terminal side lies in quadrant I. Our mission, should we choose to accept it, is to find the value of sin(θ). This is a super common type of problem, so understanding the steps will be a total game-changer for your trig adventures. We'll break it down into easy-to-follow steps. First things first, remember that the cosine function, which is the adjacent side divided by the hypotenuse in a right-angled triangle. And sine, is the opposite side over the hypotenuse. Understanding this relationship is really key to unlock the problem.

The Relationship Between Sine, Cosine and the Pythagorean Identity

Okay, so the core concept we'll use here is the Pythagorean identity: sin²(θ) + cos²(θ) = 1. This identity is a fundamental relationship in trigonometry, and it's derived directly from the Pythagorean theorem (a² + b² = c²) applied to a right triangle. This identity holds true for any angle θ. Seriously, any angle! The cool thing about this identity is that if we know either sin(θ) or cos(θ), we can solve for the other. So, since we're given cos(θ), we can totally use this identity to find sin(θ). The Pythagorean identity is your best friend in situations like these.

So, let's plug in the value of cos(θ) that we know: cos(θ) = 13/85. We'll substitute this into the Pythagorean identity. This gives us: sin²(θ) + (13/85)² = 1. Now we just need to do some algebra to isolate sin(θ). This involves squaring the fraction, subtracting the result from 1, and then taking the square root. Don't worry, it's not as scary as it sounds. We're going to use algebra to find the value of sin(θ) step-by-step. Remember, the goal is to get sin(θ) all by itself on one side of the equation.

Let's keep going. We've got our equation sin²(θ) + (13/85)² = 1. The next step is to calculate (13/85)². When you square 13/85, you'll get 169/7225. So, our equation now looks like this: sin²(θ) + 169/7225 = 1. Now, we want to isolate sin²(θ). To do that, we'll subtract 169/7225 from both sides of the equation. This gives us: sin²(θ) = 1 - 169/7225. Let's simplify the right side of the equation. To subtract the fraction, we need a common denominator, which in this case is 7225. So, 1 can be rewritten as 7225/7225. Therefore, sin²(θ) = 7225/7225 - 169/7225. Now, subtract the numerators: 7225 - 169 = 7056. Our equation now looks like this: sin²(θ) = 7056/7225. Nice!

Determining the Sign of Sin(θ) Using Quadrant Information

We're in the home stretch, people! We have sin²(θ) = 7056/7225. But we want to find sin(θ), not sin²(θ). So what do we do? We take the square root of both sides of the equation. This gives us: sin(θ) = ±√(7056/7225). Taking the square root of a fraction is the same as taking the square root of the numerator and the denominator separately. The square root of 7056 is 84, and the square root of 7225 is 85. So, sin(θ) = ±84/85. Wait a minute, we have two possible answers! This is where the quadrant information comes into play. We are told that the terminal side of θ is in quadrant I. In quadrant I, both sine and cosine are positive. That’s the key here. So, since sine must be positive in quadrant I, we choose the positive value. Therefore, sin(θ) = 84/85. We've found our answer!

The Final Answer and a Recap

So, to recap, the value of sin(θ) when cos(θ) = 13/85 and θ is in quadrant I is 84/85. We started with the Pythagorean identity, plugged in the known value of cos(θ), used some algebra to isolate sin(θ), and then considered the quadrant to determine the correct sign. This whole process showcases how different trigonometric concepts work together. Knowing the Pythagorean identity, understanding the signs of trig functions in different quadrants, and being comfortable with basic algebra are super important. Well done, you guys! You've successfully navigated a trigonometry problem and now you know how to find the sin of the angle. Remember this process, practice a few more examples, and you'll be acing these problems in no time. Keep up the awesome work!

More Examples and Practice

Example 1: Finding sin(θ) When cos(θ) is Negative

Let's switch things up a bit. Suppose we are given cos(θ) = -3/5, and we know that θ is in quadrant II. How do we find sin(θ)? The process is very similar, but we'll need to pay close attention to the signs. We again start with the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Substitute cos(θ) = -3/5: sin²(θ) + (-3/5)² = 1. Simplify: sin²(θ) + 9/25 = 1. Subtract 9/25 from both sides: sin²(θ) = 1 - 9/25. sin²(θ) = 16/25. Now, take the square root of both sides: sin(θ) = ±4/5. Because θ is in quadrant II, where sine is positive, we choose the positive value. Therefore, sin(θ) = 4/5.

This is why understanding quadrants is extremely important. The signs of sine, cosine, and tangent change depending on which quadrant your angle's terminal side is in. By understanding the signs of trigonometric functions in each quadrant, you can correctly determine the sign of the answer. Here's a quick reminder: in quadrant I, all trigonometric functions (sine, cosine, and tangent) are positive; in quadrant II, only sine is positive; in quadrant III, only tangent is positive; and in quadrant IV, only cosine is positive. The saying