Problem 67
Question
use reference angles to find the exact value of each expression. Do not use a calculator. $$ \sin \frac{2 \pi}{3} $$
Step-by-Step Solution
Verified Answer
The exact value of \(\sin \frac{2 \pi}{3}\) is \(\frac{\sqrt{3}}{2}\).
1Step 1: Identify Reference Angle
Identify the reference angle for \(\frac{2 \pi}{3}\). As one full circle is \(2 \pi\), and a straight line forms \(\pi\), the reference angle corresponds to \(\frac{\pi}{3}\), which is 60 degrees.
2Step 2: Determine the Value of Sine
The next step is to determine the value of sine for the reference angle. From the common values known, \(\sin 60^{\circ}\) (or \(\sin \frac{\pi}{3}\)) is \(\frac{\sqrt{3}}{2}\).
3Step 3: Determine the Quadrant
Since the angle \(\frac{2 \pi}{3}\) falls in the second quadrant in the unit circle where the sine function is positive, the original value should be \(\frac{\sqrt{3}}{2}\).
Key Concepts
Trigonometric FunctionsUnit CircleExact Trigonometric ValuesRadians and Degrees
Trigonometric Functions
Trigonometric functions are the fundamental building blocks of trigonometry, a branch of mathematics focused on the relationships between angles and sides of triangles, particularly right triangles. The primary trigonometric functions are sine (\textbf{sin}), cosine (\textbf{cos}), and tangent (\textbf{tan}), each of which ratios different sides of a right triangle relative to one of its acute angles.
For instance, the sine function relates the length of the side opposite to the angle to the length of the hypotenuse. In our case, finding the exact value of \( \sin \frac{2 \pi}{3} \) involves understanding these relationships. While finding this value, we look at the unit circle to determine where the angle lands and recall that in the second quadrant, sine is positive; thus, the sine of our reference angle \( \frac{\pi}{3} \) remains positive.
For instance, the sine function relates the length of the side opposite to the angle to the length of the hypotenuse. In our case, finding the exact value of \( \sin \frac{2 \pi}{3} \) involves understanding these relationships. While finding this value, we look at the unit circle to determine where the angle lands and recall that in the second quadrant, sine is positive; thus, the sine of our reference angle \( \frac{\pi}{3} \) remains positive.
Unit Circle
The unit circle is a powerful tool used in trigonometry to understand the trigonometric functions. It's a circle with a radius of 1 unit centered at the origin of a coordinate plane. Each point on the unit circle corresponds to a coordinate \( (x,y) \) that represents the cosine and sine of an angle, respectively.
Angles on the unit circle can be measured in radians, where \( 2 \pi \) radians represent a full 360-degree rotation. When dealing with \( \sin \frac{2 \pi}{3} \) as in our exercise, we locate this angle on the unit circle to help us determine the quadrant it resides in—this time, the second quadrant. Here, we leverage this tool to note that the sine function represents the y-coordinate, which is positive in both the first and second quadrants.
Angles on the unit circle can be measured in radians, where \( 2 \pi \) radians represent a full 360-degree rotation. When dealing with \( \sin \frac{2 \pi}{3} \) as in our exercise, we locate this angle on the unit circle to help us determine the quadrant it resides in—this time, the second quadrant. Here, we leverage this tool to note that the sine function represents the y-coordinate, which is positive in both the first and second quadrants.
Exact Trigonometric Values
Exact trigonometric values are specific values that trigonometric functions take at certain standard angles. These values can be known without requiring a calculator by defaulting to known triangles, such as the 30-60-90 triangle or the 45-45-90 triangle, or using the unit circle.
Common Exact Values
- \( \sin 30^\circ = \frac{1}{2} \)
- \( \sin 45^\circ = \frac{\sqrt{2}}{2} \)
- \( \sin 60^\circ = \frac{\sqrt{3}}{2} \)
Radians and Degrees
In trigonometry, angles can be measured in two units: degrees and radians. Degrees are traditionally used in everyday applications and are denoted by the symbol \( ^\circ \). There are 360 degrees in a full rotation. Radians, on the other hand, provide a direct link to the arc length on a circle and are more common in higher mathematics.
The key relationship between these units is that \( 180^\circ \) is equivalent to \( \pi \) radians. Therefore, to convert from degrees to radians, we use the equivalence \( \frac{\pi}{180^\circ} \) and vice versa. For the problem \( \sin \frac{2 \pi}{3} \) we first recognize that \( \frac{2 \pi}{3} \) radians is the same as \( 120^\circ \), allowing us to frame the problem in whichever unit is more convenient or familiar for our understanding.
The key relationship between these units is that \( 180^\circ \) is equivalent to \( \pi \) radians. Therefore, to convert from degrees to radians, we use the equivalence \( \frac{\pi}{180^\circ} \) and vice versa. For the problem \( \sin \frac{2 \pi}{3} \) we first recognize that \( \frac{2 \pi}{3} \) radians is the same as \( 120^\circ \), allowing us to frame the problem in whichever unit is more convenient or familiar for our understanding.
Other exercises in this chapter
Problem 67
Find the exact value of each expression. Do not use a calculator. $$ \csc 37^{\circ} \sec 53^{\circ}-\tan 53^{\circ} \cot 37^{\circ} $$
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Use a right triangle to write each expression as an algebraic expression. Assume that \(x\) is positive and that the given inverse trigonometric function is def
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Find a positive angle less than \(360^{\circ}\) or \(2 \pi\) that is coterminal with the given angle. $$ -\frac{\pi}{40} $$
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