Problem 63 - Algebra 2 and Trigonometry

Question

The terminal side of an angle in standard position intersects the unit circle at the point \((0.28,-0.96) .\) a. In what quadrant does the terminal side of the angle lie? b. Find, to the nearest degree, the smallest positive measure of the angle.

Step-by-Step Solution

Answer

a. Fourth Quadrant; b. 286°.

Step 1: Determine the Quadrant

The given point on the unit circle is \(0.28, -0.96\). In the Cartesian coordinate system, the x-coordinate (\(0.28\)) is positive and the y-coordinate (\(-0.96\)) is negative. This indicates that the point is located in the Fourth Quadrant, where x is positive and y is negative.

Step 2: Calculate the Angle Using Inverse Sine

The y-coordinate of the point on the unit circle corresponds to the sine of the angle. Therefore, we will use the inverse sine function to find the angle measure. Convert the y-coordinate into an angle using arcsine:\[\theta = \arcsin(-0.96)\]Using a calculator, we find that \(\theta \approx -73.74^\circ\) (since sine is negative in the fourth quadrant).

Step 3: Determine the Smallest Positive Angle

The reference angle obtained is \(73.74^\circ\), but as measured in the standard position (counterclockwise from the positive x-axis), the angle in the Fourth Quadrant needs to be calculated as:\[\text{Smallest Positive Angle} = 360^\circ - 73.74^\circ\]This results in \(\theta = 286.26^\circ\). Rounding to the nearest degree gives us \(\theta = 286^\circ\).

Key Concepts

Quadrant

In the coordinate system, every angle in standard position will fall into one of the four quadrants. A quadrant helps us understand the positioning of angles in connection to the unit circle. Angles in standard position have their vertex at the origin, and their initial side along the positive x-axis.
Identifying the quadrant of an angle is based on the signs of its x and y coordinates:

• First Quadrant: Both x and y are positive.
• Second Quadrant: x is negative, y is positive.
• Third Quadrant: Both x and y are negative.
• Fourth Quadrant: x is positive, y is negative.

In our exercise, the coordinates given are \(0.28, -0.96\). Since x is positive and y is negative, this point lies in the Fourth Quadrant. Understanding which quadrant an angle is in is crucial for determining its trigonometric properties and calculating angles accurately.

Inverse Sine

The inverse sine function, denoted as \(\arcsin\), allows us to calculate the angle whose sine value is known. Sine is associated with the y-coordinate on the unit circle. Thus, given coordinates like \(0.28, -0.96\), the y-value \((-0.96)\) represents the sine of an angle.
The inverse sine helps find the angle whose sine is \(-0.96\):\[ \theta = \arcsin(-0.96) \]This calculation assumes the angle could be negative or positive. However, it dedicates attention to specific range outputs for \(\arcsin\), typically between \(-90^\circ\) and \(90^\circ\).
In this case, calculating gives approximately \(-73.74^\circ\). Such outputs interpret the reference angle in the Fourth Quadrant, reflecting that the direct angle leading to this sine value should be that of the Fourth Quadrant. This foundational understanding helps compute angles especially when linked with the unit circle and their respective quadrants.

Reference Angle

A reference angle is essentially the smallest angle formed with the terminal side of the given angle and the x-axis. This angle is always positive, aiding in recalculating for other angles in different quadrants.
Typically, to find a reference angle in the Fourth Quadrant, you deduct the angle from \(360^\circ\). For our steps:

• The calculated inverse sine angle was \(-73.74^\circ\). This negates to a Fourth Quadrant angle reference.
• We find that the measured angle in standard form becomes:\[ \text{Smallest Positive Angle} = 360^\circ - 73.74^\circ \]
• This gives approximately \(286.26^\circ\).

Rounding this to the nearest whole number gives \(286^\circ\). This process is essential to transfer understood angle measures into standard angles, especially when determining rotations or direction on the unit circle.