Problem 14 - Algebra and Trigonometry

Question

\(9-14\) . Find the missing coordinate of \(P\) , using the fact that \(P\) lies on the unit circle in the given quadrant. $$ \frac{\text { Coordinates }}{-P\left(\frac{2}{3},\right.} ) \frac{\text { Quadrant }}{\text { II }} $$

Step-by-Step Solution

Answer

The missing coordinate is \(y = \frac{\sqrt{5}}{3}\).

Step 1: Recall Unit Circle Properties

The unit circle has a radius of 1, and any point \((x, y)\) on the unit circle satisfies the equation \(x^2 + y^2 = 1\). Since point \(P\) is on the unit circle, we know \(\left(\frac{2}{3}\right)^2 + y^2 = 1\).

Step 2: Calculate \(x^2\)

We are given \(x = \frac{2}{3}\). Calculate \(x^2\) as follows: \(\left(\frac{2}{3}\right)^2 = \frac{4}{9}\).

Step 3: Substitute and Solve for \(y^2\)

Substitute \(x^2 = \frac{4}{9}\) into the unit circle equation: \(\frac{4}{9} + y^2 = 1\). Subtract \(\frac{4}{9}\) from both sides to solve for \(y^2\): \(y^2 = 1 - \frac{4}{9} = \frac{5}{9}\).

Step 4: Solve for \(y\)

Take the square root of both sides to find \(y\): \(y = \pm \sqrt{\frac{5}{9}} = \pm \frac{\sqrt{5}}{3}\).

Step 5: Determine the Correct Sign for \(y\)

Since point \(P\) is in the second quadrant, \(x\) is negative and \(y\) is positive. Therefore, \(y = \frac{\sqrt{5}}{3}\).

Key Concepts

Quadrants

In coordinate geometry, the Cartesian plane is divided into four quadrants by the x-axis and y-axis. Each quadrant has distinct signs for coordinates:

• Quadrant I: Both x and y coordinates 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.

Understanding which quadrant a point lies in helps determine the sign of the unknown coordinate. In the exercise, point

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to represent geometric shapes, lines, and points using numbers. It gives us a powerful way to mathematically analyze shapes and solve problems.

• Points are represented as \(x, y\) coordinates.
• The distance between points, midpoints, and slopes are all fundamental in coordinate geometry.
• For circles, the equation \(x^2 + y^2 = r^2\) is used, where \(r\) is the radius.

For a unit circle, the radius is 1, therefore the equation simplifies to \(x^2 + y^2 = 1\). This equation can help find unknown coordinates if one coordinate is given.

Trigonometry

Trigonometry explores the relationships between the angles and sides of triangles and is essential in understanding circular motion, waves, and oscillations. Key concepts include:

• Trigonometric Ratios: Sine, Cosine, Tangent relate angles to side lengths in right triangles.
• Unit Circle: A circle with a radius of 1, centered at the origin, where each point corresponds to an angle and its trigonometric ratios.
• Angles: Measured in degrees or radians, where a full circle is \(360^\circ\) or \(2\pi\) radians.

For points on the unit circle like in the exercise, the coordinates \(x, y\) are equivalent to \((\cos(\theta), \sin(\theta))\), connecting the unit circle to trigonometric functions. This link aids in solving for missing coordinates.