Problem 10 - Algebra 2 and Trigonometry

Question

In \(3-11, P\) is the point at which the terminal side of an angle in standard position intersects the unit circle. The measure of the angle is \(\theta .\) For each point \(P\) the \(x\) -coordinate and the quadrant is given. Find: a. the \(y\) -coordinate of \(P\) b. \(\cos \theta\) c. \(\sin \theta\) d. \(\tan \theta\) \(\left(\frac{1}{5}, y\right),\) fourth quadrant

Step-by-Step Solution

Answer

The y-coordinate is \(-\frac{\sqrt{24}}{5}\), \(\cos \theta = \frac{1}{5}\), \(\sin \theta = -\frac{\sqrt{24}}{5}\), and \(\tan \theta = -\sqrt{24}\).

Step 1: Understand the Position of P on the Unit Circle

In a unit circle, any point \( P(x, y) \) satisfies the equation \( x^2 + y^2 = 1 \). Given that \( x = \frac{1}{5} \) and \( P \) lies in the fourth quadrant, we need to remember that in the fourth quadrant, the \( y \)-coordinate will be negative.

Step 2: Find the y-coordinate of P

Since \( x = \frac{1}{5} \), substitute it into the unit circle equation: \( \left( \frac{1}{5} \right)^2 + y^2 = 1 \). Compute \( \frac{1}{5}^2 = \frac{1}{25} \) and substitute: \( \frac{1}{25} + y^2 = 1 \). Rearrange to solve for \( y^2 \):\[ y^2 = 1 - \frac{1}{25} \]\[ y^2 = \frac{25}{25} - \frac{1}{25} = \frac{24}{25} \]Take the square root: \( y = \pm \sqrt{\frac{24}{25}} = \pm \frac{\sqrt{24}}{5} \). Since the point is in the fourth quadrant, \( y \) is negative: \( y = -\frac{\sqrt{24}}{5} \).

Step 3: Identify Cosine and Sine Values

The point \( (x, y) = \left( \frac{1}{5}, -\frac{\sqrt{24}}{5} \right) \) tells us \( \cos \theta = \frac{1}{5} \) and \( \sin \theta = -\frac{\sqrt{24}}{5} \). These represent the \( x \) and \( y \) coordinates on the unit circle, respectively.

Step 4: Calculate the Tangent of the Angle

The tangent of the angle \( \theta \) is a ratio of sine to cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Thus, we have:\[ \tan \theta = \frac{-\frac{\sqrt{24}}{5}}{\frac{1}{5}} = -\sqrt{24} \].

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are foundational in trigonometry and commonly used in various fields.

• Sine (\( \sin \theta \)): The ratio of the length of the opposite side to the hypotenuse in a right triangle. In a unit circle, it's the y-coordinate of a point.
• Cosine (\( \cos \theta \)): The ratio of the length of the adjacent side to the hypotenuse. In a unit circle, it's the x-coordinate of a point.
• Tangent (\( \tan \theta \)): The ratio of the sine to the cosine, or the opposite side to the adjacent side. On the unit circle, it's the slope of the line through the radius.

These functions help determine various properties of angles and are useful for solving trigonometric equations.

Quadrants

The coordinate plane is divided into four quadrants, each representing a different section of the plane based on the signs of the x and y coordinates. Quadrants are crucial in trigonometry for understanding the behavior of angles and functions in different parts of the unit circle.

• First Quadrant: Both x and y coordinates are positive. Angles range from 0 to 90 degrees.
• Second Quadrant: x is negative, y is positive. Angles range from 90 to 180 degrees.
• Third Quadrant: Both x and y coordinates are negative. Angles range from 180 to 270 degrees.
• Fourth Quadrant: x is positive, y is negative. Angles range from 270 to 360 degrees.

In this exercise, point P lies in the fourth quadrant, meaning the x-coordinate is positive and the y-coordinate is negative.

Angle in Standard Position

An angle in standard position is one where its vertex is at the origin of the coordinate plane, and its initial side lies along the positive x-axis. The position of the terminal side determines the quadrant in which the angle is located.For any angle in standard position:

• Vertex: Always at the origin \((0, 0)\).
• Initial Side: Lies along the positive x-axis.
• Terminal Side: Rotates from the initial side to a specific point, like point P in this exercise.

This setup enables us to easily determine the sine, cosine, and tangent of the angle based on the coordinates of the point where the terminal side intersects the unit circle.

Tangent Calculation

Calculating the tangent of an angle uses the ratio of its sine and cosine values. The formula for tangent is particularly straightforward on the unit circle, where the hypotenuse equals 1.Given that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), you can determine tangent using:

• Sine Value (\( \sin \theta \)): Corresponds to the y-coordinate of the point in the unit circle.
• Cosine Value (\( \cos \theta \)): Corresponds to the x-coordinate of the point in the unit circle.

In this exercise, the point P is \((\frac{1}{5}, -\frac{\sqrt{24}}{5})\), making the tangent:\[\tan \theta = \frac{-\frac{\sqrt{24}}{5}}{\frac{1}{5}} = -\sqrt{24}\]The tangent value helps you understand the slope of the angle relative to the x-axis, showing how steeply the line through the origin ascends or descends based on this angle.