Problem 62 - Algebra 2 and Trigonometry

Question

The terminal side of an angle in standard position intersects the unit circle at the point \((-0.8,0.6) .\) 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. Second quadrant. b. Angle is approximately 143°.

Step 1: Determine the Quadrant

To determine the quadrant where the point (-0.8, 0.6) is located, we analyze the signs of the coordinates. Since the x-coordinate is negative and the y-coordinate is positive, the point lies in the second quadrant.

Step 2: Calculate the Reference Angle Using Inverse Sine

Given that the terminal point is on the unit circle at (-0.8, 0.6), we use the y-coordinate \[\sin(\theta) = 0.6\]Use the inverse sine function to find the reference angle:\[\theta = \arcsin(0.6)\approx 36.87^\circ\]Since we work with degrees, proceed to the nearest degree, which is 37° as a reference angle.

Step 3: Determine the Actual Angle in the Second Quadrant

In the second quadrant, the angle is determined by subtracting the reference angle from 180°:\[\theta = 180^\circ - 37^\circ = 143^\circ\]

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of one that is centered at the origin of a coordinate plane.
The equation for the unit circle is \( x^2 + y^2 = 1 \). Every point on this circle corresponds to a certain angle measured from the positive x-axis.

• The full circle represents an angle of 360 degrees or \(2\pi\) radians.
• The x-coordinate of a point on the unit circle represents the cosine of the angle.
• The y-coordinate represents the sine of the angle.

The unit circle is helpful for visualizing angles and understanding the properties of sine and cosine functions, especially when combined with ideas like quadrants and reference angles.

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find the angle that corresponds to a given trigonometric value.
For example, if you know the sine of an angle is 0.6, you can use the inverse sine function, denoted as \(\arcsin\), to determine the angle itself.

• \( \arcsin \) finds an angle where the sine is a given value.
• \( \arccos \) finds an angle where the cosine is a given value.
• Inverse functions help us go from a trigonometric value back to an angle.

These functions are essential in solving problems where an angle needs to be found from known trigonometric values.

Quadrants

The coordinate plane is divided into four quadrants to help locate points, especially when working with angles. Each quadrant has specific characteristics based on the signs of the x and y coordinates.

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

In trigonometry, the quadrant determines which trigonometric functions are positive or negative. This is crucial when calculating actual angles from reference angles.

Reference Angle

A reference angle is the smallest angle between the terminal side of an angle and the x-axis. It is always a positive acute angle, less than 90°.
To find the reference angle:

• In the first quadrant, the reference angle is the angle itself.
• In the second quadrant, subtract the angle from 180°.
• In the third quadrant, subtract 180° from the angle.
• In the fourth quadrant, subtract the angle from 360°.

Understanding reference angles greatly simplifies the process of finding various trigonometric values and solving trigonometric equations, especially when dealing with angles beyond 90°.