Signs Of Coordinates For Angle 3π/4 In Standard Position

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In trigonometry, understanding the signs of coordinates in different quadrants is crucial for solving various problems. When dealing with angles in standard position, the terminal side's location determines whether the x and y coordinates are positive or negative. This article delves into the specific case of the angle θ = 3π/4, explaining why the x-coordinate is negative and the y-coordinate is positive.

Understanding Standard Position and the Unit Circle

Before diving into the specifics of θ = 3π/4, let's establish a foundation in standard position and the unit circle. An angle in standard position is an angle whose initial side lies on the positive x-axis and whose vertex is at the origin (0, 0). The terminal side is the ray that rotates from the initial side.

The unit circle is a circle with a radius of 1 centered at the origin. It's an invaluable tool in trigonometry because it directly relates angles to coordinates. For any point (x, y) on the unit circle that lies on the terminal side of an angle θ in standard position, the coordinates x and y represent the cosine and sine of θ, respectively. That is,

  • x = cos θ
  • y = sin θ

Understanding the unit circle allows us to visualize how the signs of x and y change as we move through different quadrants.

The unit circle is divided into four quadrants, labeled I, II, III, and IV, moving counterclockwise starting from the positive x-axis. Each quadrant has a distinct combination of signs for the x and y coordinates:

  • Quadrant I: x > 0, y > 0 (Both positive)
  • Quadrant II: x < 0, y > 0 (x is negative, y is positive)
  • Quadrant III: x < 0, y < 0 (Both negative)
  • Quadrant IV: x > 0, y < 0 (x is positive, y is negative)

These sign conventions are critical for determining the signs of trigonometric functions in different quadrants, as the signs of sine, cosine, and tangent (and their reciprocals) depend directly on the signs of x and y. For example, since cosine is represented by the x-coordinate and sine by the y-coordinate, in the first quadrant where both x and y are positive, both cosine and sine are positive. This understanding extends to all trigonometric functions and quadrants, making the unit circle a foundational tool in trigonometry.

Locating θ = 3π/4 on the Unit Circle

The angle θ = 3π/4 is a significant angle to consider because it helps illustrate the concepts of quadrants and sign conventions. To locate this angle, we need to understand how radians relate to the unit circle. A full circle is 2π radians, so π radians represent a half-circle, and π/2 radians represent a quarter-circle. Therefore, 3π/4 radians is three-quarters of the way from the positive x-axis to the negative x-axis, placing it in the second quadrant.

To visualize this on the unit circle, imagine starting at the positive x-axis (0 radians) and rotating counterclockwise. At π/2 radians, you are at the positive y-axis. Continuing to π radians, you are at the negative x-axis. The angle 3π/4 is exactly halfway between π/2 and π, firmly positioning it in the second quadrant. In this quadrant, any point (x, y) will have a negative x-coordinate and a positive y-coordinate.

Understanding the location of θ = 3π/4 is crucial because the coordinates of the point where the terminal side intersects the unit circle dictate the sine and cosine values of the angle. The x-coordinate corresponds to the cosine of the angle, and the y-coordinate corresponds to the sine. Since 3π/4 lies in the second quadrant, we know immediately that cos(3π/4) will be negative and sin(3π/4) will be positive. This principle applies broadly to all angles and quadrants, making it a fundamental concept in trigonometry.

By locating the angle θ = 3π/4, we can determine the signs of the coordinates of any point on its terminal side, providing a solid foundation for further trigonometric calculations.

Determining the Signs of x and y for θ = 3π/4

Now that we've located θ = 3π/4 in the second quadrant, we can definitively determine the signs of the x and y coordinates of any point on its terminal side. As established earlier, in the second quadrant, the x-coordinates are negative, and the y-coordinates are positive. This is because, in this quadrant, we are moving left (negative x direction) and up (positive y direction) from the origin.

Consider the point P(x, y) on the terminal side of θ = 3π/4. Since this point lies in the second quadrant, its x-coordinate must be negative (x < 0), and its y-coordinate must be positive (y > 0). This understanding is consistent with the unit circle where angles in the second quadrant always correspond to points with these sign characteristics.

To further clarify, imagine drawing a perpendicular line from point P to the x-axis. This creates a right-angled triangle with the hypotenuse being the distance from the origin to P. The base of the triangle lies along the negative x-axis, thus corresponding to a negative value. The height of the triangle is along the positive y-axis, corresponding to a positive value. The ratio of these sides, along with the hypotenuse, define the trigonometric ratios for θ = 3π/4, making the signs of x and y critical for determining the signs of these ratios.

Therefore, for the angle θ = 3π/4, the point P(x, y) on its terminal side will always have a negative x-coordinate and a positive y-coordinate. This result is not only a key insight for understanding this specific angle but also a general principle applicable to any angle in the second quadrant.

Significance of the Signs in Trigonometric Functions

The signs of the x and y coordinates for θ = 3π/4, and indeed for any angle, play a crucial role in determining the signs of the trigonometric functions. Recall that on the unit circle:

  • cos θ = x
  • sin θ = y
  • tan θ = y/x

Since x is negative and y is positive for θ = 3π/4, we can deduce the following:

  • cos(3π/4) is negative (because x is negative).
  • sin(3π/4) is positive (because y is positive).
  • tan(3π/4) is negative (because y/x is positive divided by negative).

These sign determinations are vital in various trigonometric applications, from solving equations to analyzing graphs. The signs of the trigonometric functions in each quadrant are often summarized using acronyms like