Problem 19 - Algebra 2 and Trigonometry

Question

In \(15-22\) , for each given angle in standard position, determine to the nearest tenth the coordinates of the point where the terminal side intersects the unit circle. $$ 229^{\circ} $$

Step-by-Step Solution

Answer

The coordinates are approximately \((-0.6, -0.8)\).

Step 1: Understanding the Problem

We need to find the point on the unit circle where the terminal side of a 229° angle in standard position intersects. Since the unit circle has a radius of 1, any point on it can be represented as \((\cos(\theta), \sin(\theta))\), where \(\theta\) is the angle in radians or degrees.

Step 2: Converting Angle to Radians

To use trigonometric functions conveniently, we should convert the angle from degrees to radians. The formula is \(\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}\). For \(229^{\circ}\), convert to radians: \[\theta_{rad} = 229 \times \frac{\pi}{180} = \frac{229\pi}{180} \approx 4.0\text{ rad}\]

Step 3: Calculating Cosine and Sine

Use a calculator to find \(\cos(229^{\circ})\) and \(\sin(229^{\circ})\). These values represent the \(x\)- and \(y\)-coordinates of the intersecting point on the unit circle. - \(\cos(229^{\circ}) \approx -0.6428\) - \(\sin(229^{\circ}) \approx -0.7660\)

Step 4: Rounding the Coordinates

Round both calculated cosine and sine values to the nearest tenth to give the final coordinates of the intersection point. - Rounded \(\cos(229^{\circ}) \approx -0.6\) - Rounded \(\sin(229^{\circ}) \approx -0.8\)

Step 5: Concluding the Coordinates

The coordinates of the point where the terminal side of the 229° angle intersects the unit circle, rounded to the nearest tenth, are \((-0.6, -0.8)\).

Key Concepts

Angle in Standard Position

An angle in standard position is a term often used in trigonometry and geometry. It refers to an angle that is positioned in a specific way on a coordinate plane. The key characteristics of an angle in standard position include:

• The vertex of the angle is located at the origin of the coordinate plane, which is the point \(0, 0\).
• The initial side of the angle starts along the positive x-axis.
• As the angle increases, the terminal side rotates counterclockwise from the initial side.

Understanding angles in standard position is essential for working with trigonometric functions and the unit circle. When the terminal side of an angle intersects the unit circle, it helps us determine the coordinates based on the trigonometric calculations. In our example with a \(229^\circ\) angle, this angle measures the counterclockwise rotation from the positive x-axis.

Trigonometric Functions

Trigonometric functions, specifically sine \(\sin\) and cosine \(\cos\), are fundamental in finding coordinates on the unit circle. They relate the angles to points on the unit circle:

• For an angle \(\theta\), the \(\cos(\theta)\) provides the x-coordinate.
• The \(\sin(\theta)\) gives the y-coordinate.

These functions rely on the radius of the unit circle, which is always 1. So any point \( (x, y) \) on the circle can be described by \( (\cos(\theta), \sin(\theta)) \). This is particularly useful because:

• When multiplying by 1, the cosine and sine values directly translate to the coordinates.
• Knowing \(\cos(\theta)\) and \(\sin(\theta)\) provides exact points along the perimeter of the circle corresponding to any angle, such as our example, \(229^\circ\).

Degree to Radian Conversion

Converting degrees to radians is necessary for many mathematical calculations, especially in trigonometry, as radians are the standard unit of angular measure in mathematics. The conversion formula is:\[ \theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180} \]This formula helps in transitioning between the two units, ensuring fluid calculations across different mathematical contexts. For the angle \(229^\circ\), converting to radians gives you:\[ \theta_{\text{rad}} = 229 \times \frac{\pi}{180} \approx 4.0 \text{ rad} \]Radian measures are advantageous in certain calculus computations and when working with circular movements since they relate more directly to arc lengths on the unit circle. This ease of use explains why many mathematical problems, including trigonometry, prefer radians.

Coordinates on the Unit Circle

The concept of coordinates on the unit circle stems from the circle's unique radius of 1. Each point on the unit circle corresponds to an angle \(\theta\) and can be described as \((\cos(\theta), \sin(\theta))\). For a given angle in standard position, its terminal side's intersection with the unit circle gives these specific coordinates.

• When addressing the angle \(229^\circ\), utilizing trigonometric functions, we find the point \((-0.6428, -0.7660)\) before rounding.
• After rounding to the nearest tenth, the coordinates become \((-0.6, -0.8)\).

These coordinates are crucial, as they inform us about the precise location along the unit circle where the terminal side meets the circle. Such points are useful for various applications in both theoretical and applied mathematics, such as wave functions, oscillations, and circular motion.