Sine, Cosine, Obtuse Angles: A Trig Exploration

Hey guys! Today, we're diving deep into the fascinating world of trigonometry, specifically focusing on how to tackle problems involving sine and cosine functions when dealing with obtuse angles. We've got a juicy problem to dissect: given that sin(x) = 4/8 and cos(y) = -1/1, where x is an obtuse angle and 180° < y, we're going to explore the ins and outs of finding these angles and understanding the implications of their positions on the unit circle. This isn't just about crunching numbers; it's about grasping the underlying concepts that make trigonometry so powerful. So, buckle up and let's get started!

Before we jump into the nitty-gritty, let's refresh our understanding of the fundamental trigonometric functions: sine and cosine. These functions are the cornerstones of trigonometry, and their behavior is best visualized using the unit circle. The unit circle, with a radius of 1, provides a geometric interpretation of trigonometric functions. Imagine a point moving around this circle; its x-coordinate represents the cosine of the angle formed with the positive x-axis, and its y-coordinate represents the sine of the angle.

The sine of an angle, often abbreviated as sin, corresponds to the y-coordinate of the point where the terminal side of the angle intersects the unit circle. In simpler terms, it tells us how high or low the point is on the circle. The cosine of an angle, abbreviated as cos, corresponds to the x-coordinate of that same point. It tells us how far left or right the point is.

The unit circle is divided into four quadrants, each spanning 90 degrees. This division is crucial because the signs of sine and cosine vary across these quadrants. In the first quadrant (0° to 90°), both sine and cosine are positive. In the second quadrant (90° to 180°), sine is positive, but cosine is negative. In the third quadrant (180° to 270°), both sine and cosine are negative. Finally, in the fourth quadrant (270° to 360°), sine is negative, while cosine is positive. This quadrant behavior is essential for solving trigonometric problems, especially when dealing with angles beyond the first quadrant.

Now, let's zoom in on obtuse angles. An obtuse angle is an angle that measures greater than 90° but less than 180°. This places it squarely in the second quadrant of the unit circle. A key characteristic of angles in this quadrant is that their sine values are positive, while their cosine values are negative. This is because, as we discussed earlier, the y-coordinate (sine) is positive in the second quadrant, and the x-coordinate (cosine) is negative.

When we're given the sine of an obtuse angle, like sin(x) = 4/8 in our problem, we know that x lies in the second quadrant. However, there's a catch! The sine function is also positive in the first quadrant. So, we need to use the information about x being obtuse to pinpoint its exact location. The cosine value, being negative in the second quadrant, further confirms this. Understanding this concept is critical for solving problems that involve finding angles when given their trigonometric ratios.

Let's get back to our problem. We're given that sin(x) = 4/8, which simplifies to sin(x) = 1/2. We also know that x is an obtuse angle. To find the value of x, we need to think about which angles have a sine of 1/2. If we recall our special right triangles, particularly the 30-60-90 triangle, we know that sin(30°) = 1/2. However, we're dealing with an obtuse angle, which means x is not 30°. This is where the unit circle and our understanding of quadrants come into play.

The sine function is positive in both the first and second quadrants. While 30° is the reference angle in the first quadrant, there's a corresponding angle in the second quadrant with the same sine value. This angle is the supplement of 30°, which is 180° - 30° = 150°. Since 150° is an obtuse angle (between 90° and 180°), it satisfies the condition that x is obtuse. Therefore, x = 150° is the solution we're looking for.

Now, let's shift our focus to the second part of the problem: cos(y) = -1/1, which simplifies to cos(y) = -1, and 180° < y. This equation tells us that the cosine of angle y is -1 and that y is greater than 180°. To decipher this, we again turn to the unit circle. The cosine function represents the x-coordinate of a point on the unit circle. So, we need to find the angle where the x-coordinate is -1.

Looking at the unit circle, we see that the point (-1, 0) corresponds to an angle of 180°. However, our condition states that y > 180°. This means we need to consider angles greater than 180°. As we continue around the unit circle, we reach the point (-1, 0) again at 360°, but since the cosine function has a period of 360°, we need to think about the implications of y > 180°. Since cos(180°) = -1 and we need an angle greater than 180°, the most direct solution within the standard 0° to 360° range is y = 180°. However, the problem statement