Understanding Signs Of X And Y At 3π/4 Angle

by Alex Johnson 45 views

Hey there, math enthusiasts! Let's dive into a neat little geometry problem. We're going to explore what happens when a point P(x, y) sits on the terminal side of an angle, specifically when that angle (θ) is equal to 3π/4 radians, positioned in the standard way. Our main goal is to figure out the signs of the x and y values. Are they both positive, both negative, or a mix? Let’s find out!

The Basics of Angles and Coordinates

First things first, let's get our bearings. When we talk about an angle in standard position in the coordinate plane, we mean the angle's starting side (initial side) is on the positive x-axis, and it rotates counter-clockwise. Think of it like a spinning ray starting from the right and moving upwards. The terminal side is the ray's final position after the rotation. The point P(x, y) is just a point somewhere along that terminal side. The values x and y represent the point's coordinates, telling us how far right/left (x) and up/down (y) the point is from the origin (0, 0).

Now, the angle 3π/4 radians is a specific rotation. Remember, a full circle is 2π radians. Half a circle is π radians. Our angle, 3π/4, is a little less than a full half-circle (π). It’s in the second quadrant. This position is really important to know where our point P(x, y) will be located. Understanding this sets the foundation for our next steps in finding the signs of the coordinates.

Now to determine the signs of x and y for the point P(x, y) that lies on the terminal side of an angle of θ = 3π/4 in standard position.

Visualizing the 3π/4 Angle

Let’s picture this on a graph. Imagine the x and y axes intersecting at the origin. The angle 3π/4 is a special angle – it’s a 135-degree angle in degrees (because 3π/4 radians * (180/π) = 135 degrees). This means if we draw a line from the origin at this angle, it will sit in the second quadrant. The second quadrant is the top-left section of our graph. Think of dividing a pizza into four slices; we are in the second slice.

So, any point P(x, y) on the terminal side of this angle must be in the second quadrant. Since the angle is 135 degrees, and the x-axis marks 0 degrees, this means the angle is more than 90 degrees but less than 180 degrees. This provides an excellent visual of what we are dealing with. Visualize the angle and the placement of the point, this is super important!

Let's get even more detailed. Any point on the terminal side of the 3π/4 angle will have an x-coordinate that is negative. The reason is simple, the x values are negative to the left of the origin (0, 0). Likewise, the y coordinates are always positive in the second quadrant, they are above the x-axis. Thus, for any point P(x, y), x will be negative, and y will be positive. This will help us solve the main question.

Determining the Signs of x and y

Alright, let’s get down to the core of the problem: what are the signs of x and y? Think about how the x and y coordinates behave across the quadrants.

  • First Quadrant: Both x and y are positive. This is the top-right section of our graph.
  • Second Quadrant: x is negative, and y is positive. This is where our 3π/4 angle lives!
  • Third Quadrant: Both x and y are negative. This is the bottom-left section.
  • Fourth Quadrant: x is positive, and y is negative. This is the bottom-right section.

Since our angle of 3π/4 radians (135 degrees) places our point P(x, y) in the second quadrant, we know that the x-coordinate will be negative, and the y-coordinate will be positive. Therefore, the answer to the question must be that x is negative, and y is positive.

Now, if the angle were, say, 7π/4 (315 degrees), the terminal side would be in the fourth quadrant. In this case, x would be positive, and y would be negative. The signs of x and y are always determined by the quadrant the terminal side of the angle falls into.

Putting it All Together

So, to recap, when the point P(x, y) lies on the terminal side of an angle θ = 3π/4 in standard position: The x-coordinate is negative because the point lies to the left of the y-axis, and the y-coordinate is positive because the point lies above the x-axis. Remember that the angle 3π/4 places the terminal side into the second quadrant.

Here’s the breakdown:

  • The angle is in the second quadrant.
  • x is negative.
  • y is positive.

Therefore, we have successfully determined the signs of x and y. Remember to always visualize the angle and the quadrants to help solidify your understanding.

Further Exploration and Key Takeaways

Keep in mind that the signs of x and y change depending on which quadrant the terminal side of the angle lies in. The unit circle is a fantastic tool to use for visualizing angles and their corresponding coordinate values, as is plotting the angle on a graph.

  • Quadrant I: (+, +)
  • Quadrant II: (-, +)
  • Quadrant III: (-, -)
  • Quadrant IV: (+, -)

Understanding these basic quadrant rules helps you determine the signs of the coordinates for any angle. Keep practicing with different angles and points, and you'll become a pro in no time! Remember, these concepts are foundational for trigonometry, so getting a solid grasp now will pay off handsomely in your future studies. Also, remember to draw and visualize – it will help you a lot to understand these concepts.

Let's get some practice problems to reinforce our knowledge: What if we have a point P(x, y) on the terminal side of an angle of 5π/3? In which quadrant does this angle lie, and what are the signs of x and y? Always remember the unit circle to see how the angles change through the quadrants. Understanding this topic is critical to solving many problems in mathematics and science!

Conclusion

So there you have it! We've successfully navigated the coordinate plane and determined the signs of x and y for an angle of 3π/4. Remember, it's all about understanding the quadrants and how the x and y values change within them. Keep practicing, and you'll master this concept in no time! Keep exploring and don't be afraid to ask questions; there's always more to learn and discover in the wonderful world of mathematics.

For more detailed explanations and examples, you can check out these resources:

  • Khan Academy: They offer excellent videos and practice exercises on trigonometry and the unit circle. (Khan Academy Trigonometry)