Problem 60 - Algebra 2

Question

Open-Ended Find the measures of four angles in standard position that have a sine of \(0.5 .\) . (Hint: Use the unit circle and right triangles.)

Step-by-Step Solution

Answer

The four angles in standard position that have a sine of \(0.5\) are \(\pi /6\) radians (\(30^{\circ}\)), \(5 \pi /6\) radians (\(150^{\circ}\)), \(-7\pi /6\) radians (\(-210^{\circ}\)), and \(-11\pi /6\) radians (\(-330^{\circ}\)).

Step 1: Identify points on the unit circle

The sine value of an angle corresponds to the y-coordinate on the unit circle. Therefore, we are looking for the points on the unit circle where the y-coordinate is 0.5. This occurs at two points: \((\sqrt{3}/2 , 0.5)\) and \((- \sqrt{3}/2 , 0.5)\).

Step 2: Identify corresponding angles in positive direction

Draw lines from the origin to these points on the unit circle. The angles that these lines make with the positive x-axis in the counterclockwise direction are the angles we are interested in. The first angle (for \((\sqrt{3}/2 , 0.5)\)) is \(\pi /6\) radians or \(30^{\circ}\) and the second angle (for \((- \sqrt{3}/2 , 0.5)\)) is \(5 \pi /6\) radians or \(150^{\circ}\).

Step 3: Identify corresponding angles in negative direction

Moving clockwise from the positive x-axis to the angle gives us the negative representation of the angle. The angles in the clockwise direction to the same points on the unit circle are \(-7\pi /6\) radians or \(-210^{\circ}\) and \(-11\pi /6\) radians or \(-330^{\circ}\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry and is used to explore angle measures and trigonometric functions. It is a circle with a radius of 1, centered at the origin of a coordinate plane.
On this circle, every point \(x, y\) has coordinates where \(x\) represents the cosine of the angle & \(y\) represents the sine of the angle.
Because the radius is 1, it simplifies many trigonometric calculations.

• Angles are usually measured from the positive x-axis.
• This circle helps in visualizing angles in radians and degrees.
• The unit circle is also instrumental in understanding trigonometric functions for all angle measures, including angles beyond \(360^{\circ}\) or \(2\pi\) radians.

When you plot angles moving counterclockwise around the circle, these correspond to positive angles. Clockwise movement leads to negative angles.
In problems like finding angles with a given sine value, you simply locate the y-coordinate on this circle.

Sine Function

The sine function is a crucial trigonometric function that relates the angle in a right triangle to the ratio of the opposite side over the hypotenuse.
On the unit circle, this corresponds to the y-coordinate of a point made by the angle with the x-axis.
Sine values range from \(-1\) to \(+1\), as the y-coordinates on the unit circle span this interval.- Sine of an angle \(\theta\) = y-coordinate on the unit circle.- Periodic in nature, repeating every \(2\pi\) radians or \(360^{\circ}\).For instance, if the sine of an angle is \(0.5\), these y-coordinates tell you where on the unit circle that occurs:

• At \(30^{\circ} (\pi/6)\) and the supplementary angle \(150^{\circ} (5\pi/6)\) in the positive direction.
• Additionally, you find equivalents in the negative direction by moving clockwise: \(-210^{\circ} (-7\pi/6)\) and \(-330^{\circ} (-11\pi/6)\).

Radians and Degrees

Radians and degrees are two common units for measuring angles, often used in trigonometry and geometry.
Degrees are a more traditional unit, dividing a circle into \(360\) parts. Radians provide a more natural description based on the radius of the circle.
One complete revolution around a circle equals \(360^{\circ}\) or \(2\pi\) radians.Conversion between radians and degrees:

• For degrees to radians: Multiply by \(\pi/180\).
• For radians to degrees: Multiply by \(180/\pi\).

In the context of the unit circle and trigonometric calculations, radians are often preferred because they simplify the math. For the angle measures derived, you'll see:
- \(\pi/6\) radians is equivalent to \(30^{\circ}\).- \(5\pi/6\) radians converts to \(150^{\circ}\).

• This understanding aids in calculating sines, cosines, and comprehending periodicity in trigonometric functions.

Angle Measures

Angle measures are critical in trigonometry for describing rotations and determining trigonometric function values.
When an angle is in standard position, its vertex is at the origin with the initial side on the positive x-axis.
You can describe angles using either positive or negative measures, based on the direction of rotation:

• Positive angles rotate counterclockwise.
• Negative angles rotate clockwise.

For example, finding four angles with a sine of \(0.5\) involves:

• Positive measures: \(30^{\circ} (\pi/6)\) and \(150^{\circ} (5\pi/6)\).
• Negative measures: \(-210^{\circ} (-7\pi/6)\) and \(-330^{\circ} (-11\pi/6)\).

Recognizing that angles repeat periodically also helps in identifying various equivalent angle measures for the same sine value, extending beyond \(360^{\circ}\).