Problem 8 - Algebra 2 and Trigonometry

Question

In \(3-12\) , find the exact function value of each of the following if the measure of the angle is given in radians. $$ \sin \frac{4 \pi}{3} $$

Step-by-Step Solution

Answer

\(\sin \frac{4\pi}{3} = -\frac{\sqrt{3}}{2}\)

Step 1: Convert Angle to Degrees

First, convert the angle from radians to degrees to better understand its position on the unit circle. The conversion is done using the formula: Degrees = Radians × (180/π). So, \( \frac{4\pi}{3} \times \frac{180}{\pi} = 240^{\circ} \).

Step 2: Determine Reference Angle

The angle \(240^{\circ}\) is in the third quadrant. Calculate the reference angle by finding the difference from \(180^{\circ}\): Reference angle = \(240^{\circ} - 180^{\circ} = 60^{\circ}\).

Step 3: Recall Sine of Reference Angle

Recall that the sine of the reference angle \(60^{\circ}\) is \(\sin(60^{\circ}) = \frac{\sqrt{3}}{2}\).

Step 4: Determine Sign in Third Quadrant

Since \( \frac{4\pi}{3} \) or \( 240^{\circ} \) is in the third quadrant where sine values are negative, apply the negative to the sine of the reference angle. Thus, \( \sin \frac{4\pi}{3} = -\frac{\sqrt{3}}{2} \).

Key Concepts

Radian Conversion

Understanding how to convert between radians and degrees is essential in trigonometry. Radians and degrees are two different units for measuring angles, just like meters and yards measure length.

• To convert from radians to degrees, use the formula: Degrees = Radians × (180/π).
• This conversion is helpful for understanding angles in a way that might feel more familiar, especially when working with the unit circle.

For example, to convert \( \frac{4\pi}{3} \) radians to degrees, you multiply by the conversion factor, resulting in 240 degrees. This means the angle is swept out 240 degrees from the positive x-axis, helping to locate it on the unit circle.

Unit Circle

The unit circle is a fundamental tool in trigonometry. It is a circle with a radius of one, centered at the origin of a coordinate plane.

• The unit circle allows us to define trigonometric functions such as sine, cosine, and tangent for all angles, both positive and negative.
• Each point on the unit circle corresponds to the cosine and sine of an angle, represented by \( (\cos(\theta), \sin(\theta)) \).

The angle itself is measured from the positive x-axis in a counterclockwise direction. When you converted \( \frac{4\pi}{3} \) into degrees, you found it corresponds to an angle of 240 degrees in the unit circle, placing it in the third quadrant where both basic trigonometric functions take specific signs.

Reference Angle

A reference angle helps easily calculate trigonometric values, despite the angle you are dealing with.

• A reference angle is always positive and is the smallest angle that the terminal side of a given angle makes with the x-axis.
• This is particularly useful since the sine, cosine, and tangent of angles in different quadrants can be determined using the reference angle.

For example, with the angle of 240 degrees located in the third quadrant, its reference angle can be determined by subtracting 180 degrees. So, the reference angle is 60 degrees, which helps in gathering trigonometric values based on basic known angles.

Sine Function

The sine function is one of the basic trigonometric functions.

• It gives the y-coordinate of the point on the unit circle that corresponds to an angle.
• The sine of an angle tells us the vertical position of a point on the unit circle.
• For angles in the unit circle, the sine function shows periodic and symmetric properties.

When \( \sin(240^{\circ}) \) or \( \sin\left(\frac{4\pi}{3}\right) \) is calculated, the sine value is negative because the angle is in the third quadrant. Reviewing sine at the reference angle of 60 degrees, you recall the basic fact that \( \sin(60^{\circ}) = \frac{\sqrt{3}}{2} \). Given sine's sign that is negative in the third quadrant, the result is \( \sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2} \). This reinforces how quadrant locations affect the sign of the sine function.