Question

Find the exact value of each expression. Do not use a calculator. $$ \sin \left(-\frac{2 \pi}{3}\right) $$

Short answer

\(-\frac{\sqrt{3}}{2}\)

Step-by-step solution

Identify the Angle in a Standard Position

The given angle is \(-\frac{2 \pi}{3}\). Since this is a negative angle, it means rotating \(-\frac{2 \pi}{3}\) radians in the clockwise direction.

Find the Coterminal Angle

To find a coterminal angle that is positive, add \(2 \pi\) to \(-\frac{2 \pi}{3}\): \(-\frac{2 \pi}{3} + 2 \pi = \frac{4 \pi}{3}\). This is the angle that is coterminal with \(-\frac{2 \pi}{3}\).

Determine the Reference Angle

The reference angle is the smallest angle formed between the terminal side of \(-\frac{2 \pi}{3}\) (or its coterminal angle \(\frac{4 \pi}{3}\)) and the x-axis. \(\frac{4 \pi}{3}\) is in the third quadrant, and the reference angle is calculated as \|\frac{4 \pi}{3} - \pi| = \frac{\pi}{3}\.

Evaluate the Sine of the Reference Angle

The sine of the reference angle \(\frac{\pi}{3}\) is \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\).

Determine the Sign in the Third Quadrant

In the third quadrant, sine values are negative. Hence, \(\sin\left(\frac{4 \pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right)\).

Write the Final Answer

Thus, \(\sin\left(\frac{4 \pi}{3}\right) = -\frac{\sqrt{3}}{2}\) and therefore, \(\sin\left(-\frac{2 \pi}{3}\right) = -\frac{\sqrt{3}}{2}\).

Key concepts

Sin Function

The sine function, denoted as \(\text{sin}\), is a foundational trigonometric function used to relate the angle of a right triangle to the lengths of its opposite side and hypotenuse. When given an angle \( \theta \), the value of \(\text{sin}(\theta)\) is the y-coordinate of the corresponding point on the unit circle. Specifically, for any angle \( \theta \), the function can be defined as \(\text{sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\) in a right triangle or simply the y-value in the unit circle representation. Understanding the sine function is crucial for solving various trigonometric problems, such as finding the exact value of expressions like \( \text{sin}\big(-\frac{2 \theta}{3}\big) \).

Reference Angle

A reference angle is the smallest angle that the terminal side of a given angle makes with the x-axis. It helps to simplify trigonometric calculations, as the trigonometric functions of any angle are the same as those of its reference angle, but possibly with a different sign, depending on the quadrant. For an angle \( \theta \) in standard position, if \( \theta \) lies in the second quadrant, its reference angle is found by subtracting \( \theta \) from \( 180^\text{o} \) or \( \frac{\theta}{2} \text{ radians} \). In our example of \(-\frac{2\theta}{3}\), we first find its positive coterminal angle \(\frac{4\theta}{3}\), which lies in the third quadrant. Consequently, the reference angle for \(\frac{4\theta}{3}\) is calculated as \(\big| \frac{4\theta}{3} - \theta \big| = \frac{\theta}{3}\). This reference angle helps to easily find the sine value by considering the simpler angle \(\frac{\theta}{3}\).

Coterminal Angles

Coterminal angles are angles that share the same terminal side but are measured differently, either by adding or subtracting full rotations of \(2\theta\) radians (or 360 degrees). To find a positive coterminal angle for a given angle, especially when dealing with negative angles like \(-\frac{2 \theta}{3}\), you can add \(2\theta\). This gives us \(-\frac{2 \theta}{3} + 2 \theta = \frac{4 \theta}{3}\). Understanding coterminal angles ensures that we always work with positive angles that are simpler to interpret and use for further calculations. It brings clarity, especially when determining which quadrant the angle lies in and subsequently its reference angle and corresponding trigonometric values.

Quadrants in Trigonometry

The x-y coordinate plane is divided into four quadrants, each covering a 90-degree (or \(\frac{\theta}{2}\) radian) section. Knowing in which quadrant an angle lies helps us determine the signs of trigonometric functions for that angle.
For quick reference, the quadrants are:

• First Quadrant: Angles between 0 and \(\frac{\theta}{2}\). All trigonometric values are positive here.
• Second Quadrant: Angles between \(\frac{\theta}{2}\) and \(\theta\). Only sine values are positive, while cosine and tangent are negative.
• Third Quadrant: Angles between \(\theta\) and \(\frac{3\theta}{2}\). Tangent values are positive, whereas sine and cosine are negative.
• Fourth Quadrant: Angles between \(\frac{3\theta}{2}\) and \(2\theta\). Only cosine values are positive, sine and tangent are negative.

In the example \(-\frac{2\theta}{3}\), we find it coterminal to \(\frac{4\theta}{3}\) which lies in the third quadrant. Knowing this is essential as it informs us that the sine value will be negative.