Problem 10 - Precalculus Mathematics for Calculus

Question

Find the missing coordinate of \(P,\) using the fact that \(P\) lies on the unit circle in the given quadrant. Coordinates $$P\left(\quad,-\frac{7}{25}\right)$$ Quadrant IV

Step-by-Step Solution

Answer

The missing coordinate is \(\frac{24}{25}\).

Step 1: Understanding the Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin \((0,0)\) in the coordinate plane. Any point \((x, y)\) on the unit circle satisfies the equation \(x^2 + y^2 = 1\).

Step 2: Substitute the Known Coordinates

We know that the \(y\)-coordinate of point \(P\) is \(-\frac{7}{25}\). Substituting this into the unit circle equation gives:\[x^2 + \left(-\frac{7}{25}\right)^2 = 1\]

Step 3: Calculate \(y^2\)

Calculate \(y^2\) knowing that \(y = -\frac{7}{25}\):\[\left(-\frac{7}{25}\right)^2 = \frac{49}{625}\]

Step 4: Simplify the Equation

Substitute \(y^2 = \frac{49}{625}\) back into the equation:\[x^2 + \frac{49}{625} = 1\]

Step 5: Solve for \(x^2\)

Subtract \(\frac{49}{625}\) from both sides to isolate \(x^2\):\[x^2 = 1 - \frac{49}{625}\]Calculate:\[x^2 = \frac{625}{625} - \frac{49}{625} = \frac{576}{625}\]

Step 6: Find \(x\)

Take the square root of both sides to find the value of \(x\):\[x = \pm \sqrt{\frac{576}{625}}\]\[x = \pm \frac{24}{25}\]

Step 7: Consider the Quadrant

Since point \(P\) lies in Quadrant IV, where \(x\) is positive and \(y\) is negative, choose \(x = \frac{24}{25}\).

Key Concepts

Quadrants

When we look at the plane where coordinates are mapped, it is divided into four parts called quadrants. Each quadrant has different signs for its x and y values:

• Quadrant I: Both x and y are positive.
• Quadrant II: x is negative and y is positive.
• Quadrant III: Both x and y are negative.
• Quadrant IV: x is positive and y is negative.

In the exercise, point P is in Quadrant IV. Knowing the quadrant helps decide the signs of coordinates. In Quadrant IV, x is positive while y is negative. This plays a key role when solving, as we look for positive values of x.

Cartesian coordinates

In the Cartesian coordinate system, we place any point in a space defined by two numbers, x and y, which represent its horizontal and vertical position, respectively. The coordinate plane is made up of two axes:

• The x-axis is horizontal.
• The y-axis is vertical.

The crossing point of these axes is called the origin, marked as (0, 0). Points are noted as (x, y). For example, in the exercise, we have the coordinate of point P as \(\left( \,?\,,-\frac{7}{25} \right)\) which means the y-coordinate is -\(\frac{7}{25}\), and x is to be determined.

Pythagorean identity

The Pythagorean identity is a significant principle found in trigonometry, especially on the unit circle. It says that for any point (x, y) on the unit circle,
\[x^2 + y^2 = 1\]
holds true. This reflects the Pythagorean theorem for a right-angled triangle, where 1 is the hypotenuse (radius of the unit circle) while x and y are the triangle's other two sides.
In our unit circle problem, knowing y lets us solve for x using this identity. Assuming y = -\(\frac{7}{25}\), we set up the equation and solve for x, leading us to determine \(x^2 = \frac{576}{625}\).

Coordinate geometry

Coordinate geometry, also known as analytic geometry, lets us understand geometric concepts through algebra and the coordinate system. It enables us to find distances, slopes, and angles among points in a plane. For instance, analyzing points as coordinates on graphs
has various applications such as locating points like P in the exercise.
By observing that point P lies on the unit circle, we apply coordinate geometry concepts to use x and y values in equations derived from geometric properties. This results in simplifying and solving for the missing coordinate using our equations to reflect the quadrant's sign rules in choosing the correct value of x.