Problem 23 - Algebra 2 and Trigonometry

Question

If \(P\left(\frac{2}{3}, y\right)\) is a point on the unit circle and on the terminal side of an angle in standard position with measure \(\theta,\) find: a. \(y\) b. \(\sin \theta\) c. \(\cos \theta\)

Step-by-Step Solution

Answer

\(y = \pm \frac{\sqrt{5}}{3}\); \(\sin \theta = \pm \frac{\sqrt{5}}{3}\); \(\cos \theta = \frac{2}{3}\).

Step 1: Understanding the Unit Circle

The unit circle is defined as a circle with a radius of 1. The equation of the unit circle is \(x^2 + y^2 = 1\), where \(x\) and \(y\) are the coordinates of any point on the circle.

Step 2: Identify the X-Coordinate

From the problem, we know that the point \(P\left(\frac{2}{3}, y\right)\) is on the unit circle. This means that the \(x\)-coordinate is \(\frac{2}{3}\).

Step 3: Substitute into the Unit Circle Equation

Substitute \(x = \frac{2}{3}\) into the unit circle equation: \(\left(\frac{2}{3}\right)^2 + y^2 = 1\). This simplifies to \(\frac{4}{9} + y^2 = 1\).

Step 4: Solve for \(y\)

Rearrange the equation to solve for \(y^2\): \(y^2 = 1 - \frac{4}{9}\). This simplifies to \(y^2 = \frac{5}{9}\). Taking the square root of both sides, we find \(y = \pm\frac{\sqrt{5}}{3}\).

Step 5: Determine the Value of \(y\)

Since no specific quadrant is indicated and given that the unit circle has points symmetric about the origin, both values \(\frac{\sqrt{5}}{3}\) and \(-\frac{\sqrt{5}}{3}\) can exist. However, unless specified by quadrant or additional conditions, we maintain both options for \(y\).

Step 6: Identify \(\sin \theta\) and \(\cos \theta\)

For a point \((x, y)\) on the unit circle, \(\cos \theta = x\) and \(\sin \theta = y\). Therefore, \(\cos \theta = \frac{2}{3}\) and \(\sin \theta = \pm \frac{\sqrt{5}}{3}\).

Key Concepts

Trigonometric Functions

Trigonometric functions, essential in mathematics, describe relationships in right-angled triangles. They detail how angles correlate with side lengths. On the unit circle, these functions help find

• the sine (\(\sin \theta\)) which corresponds to the y-coordinate,
• the cosine (\(\cos \theta\)) corresponding to the x-coordinate,
• and the tangent (\(\tan \theta\)), a ratio of the sine to the cosine.

Consider the point \(P\left(\frac{2}{3}, y\right)\) on the unit circle. Here, \(x = \frac{2}{3}\). This immediately tells us that \(\cos \theta = \frac{2}{3}\). For \(\sin \theta\), we solve for \(y\), and we find it as \(\pm \frac{\sqrt{5}}{3}\). These basic trigonometric functions are crucial for any problems involving angles and distances.

Coordinate Geometry

Coordinate geometry is all about the spatial arrangement of points and how they interact. Whenever we dive into unit circles or trigonometric identities, it provides a way to visualize and compute without delving into cumbersome details. The unit circle is a primary example. With its center at the origin \((0, 0)\) and radius 1, any point \((x, y)\) on this circle corresponds to a particular angle in standard position.For example, when we see a point like \(P\left(\frac{2}{3}, y\right)\), coordinate geometry guides us to understand and use its position relative to the origin. The x-coordinate represents the cosine function, while the y-coordinate reveals the sine function. This simple setup enables us to derive relations, tackle problems, and build knowledge of geometry's deeper aspects.

Pythagorean Identity

The Pythagorean identity is a foundational aspect of trigonometry, emerging directly from the equation of the unit circle: \(x^2 + y^2 = 1\). This equation ensures that for any point \((x, y)\) on the circle:

• the sum of the square of the cosine (\(\cos^2 \theta\)) and the square of the sine (\(\sin^2 \theta\)) equals 1.

In this problem, the identity helps us solve for \(y^2\). Given \(x = \frac{2}{3}\), substituting into the identity yields \(\left(\frac{2}{3}\right)^2 + y^2 = 1\), allowing us to extract \(y^2 = \frac{5}{9}\). Taking the square root gives the two possible values for \(y\) as \(+\frac{\sqrt{5}}{3}\) and \(-\frac{\sqrt{5}}{3}\). This identity is invaluable in enabling the derivation of such crucial characteristics of trigonometric functions.