Find the missing coordinate of p, using the fact that p lies on the unit circle in the given quadrant.
P( ), -3/7
do you mean P( ...., -3/7) ?
x^2 + (-3/7)^2 = 1^2
x^2 = 1 - 9/49 = 40/49
x = ±√40/7 = ± 2√10/7
we see that the y is negative, so the point could be in III or IV, thus the ±
In order to find the missing coordinate of point P, we need to use the fact that it lies on the unit circle in a given quadrant.
Since the y-coordinate is given as -3/7, we can use the Pythagorean identity to find the missing x-coordinate. The Pythagorean identity states that for any point (x, y) on the unit circle, x^2 + y^2 = 1.
Substituting the given y-coordinate, we have:
x^2 + (-3/7)^2 = 1
x^2 + 9/49 = 1
Subtracting 9/49 from both sides:
x^2 = 1 - 9/49
x^2 = 40/49
Now we take the square root of both sides to solve for x:
x = ±sqrt(40/49)
Since point P is in a specific quadrant, we need to determine the sign of the x-coordinate. Since the y-coordinate is negative (-3/7), point P must be in the third quadrant. In the third quadrant, both x and y are negative.
So, the missing coordinate of P is:
P( -sqrt(40/49), -3/7)
To find the missing coordinate of point P on the unit circle, we need to use the fact that P lies in a given quadrant and that the point is on the unit circle.
First, let's understand the concept of the unit circle. The unit circle is a circle with a radius of 1 unit, centered at the origin of the coordinate plane (0, 0). It is used in trigonometry to define the values of sine, cosine, and tangent of angles.
Given that P is on the unit circle and has a y-coordinate of -3/7, we can determine the missing x-coordinate and the quadrant of P.
Since the y-coordinate is negative, we know that P lies in either the third or fourth quadrant of the coordinate plane, as those are the quadrants where the y-coordinate is negative.
To find the x-coordinate of P, we can use the Pythagorean theorem. The Pythagorean theorem states that, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
In the unit circle, the x-coordinate represents the adjacent side of a right triangle, and the y-coordinate represents the opposite side. The hypotenuse is always 1 (as it is a unit circle).
Applying the Pythagorean theorem, we have:
1^2 = (x-coordinate)^2 + (-3/7)^2
Simplifying the equation:
1 = (x-coordinate)^2 + 9/49
Rearranging the equation:
(x-coordinate)^2 = 1 - 9/49
(x-coordinate)^2 = 40/49
Taking the square root of both sides:
x-coordinate = ±√(40/49)
Since P lies in a given quadrant, we can determine the sign of the x-coordinate based on the quadrant.
Considering P lies in either the third or fourth quadrant, both of which have negative x-coordinates, we can conclude that the missing x-coordinate is -√(40/49).
Therefore, the missing coordinate of P is:
P(-√(40/49), -3/7)