Problem 3 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function at the given real number. $$ \begin{array}{llll}{\text { (a) } \sin \frac{2 \pi}{3}} & {\text { (b) } \cos \frac{2 \pi}{3}} & {\text { (c) } \tan \frac{2 \pi}{3}}\end{array} $$

Step-by-Step Solution

Answer

(a) \(\frac{\sqrt{3}}{2}\), (b) \(-\frac{1}{2}\), (c) \(-\sqrt{3}\)

Step 1: Understand the Angle

The angle \( \frac{2\pi}{3} \) is in radians. It lies in the second quadrant of the unit circle, where sine values are positive, and cosine and tangent values are negative.

Step 2: Use Reference Angle for Sine

In the second quadrant, the reference angle for \( \frac{2\pi}{3} \) is \( \pi - \frac{2\pi}{3} = \frac{\pi}{3} \). The sine of an angle in the second quadrant is positive and equal to the sine of its reference angle. Hence, \( \sin \frac{2\pi}{3} = \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \).

Step 3: Use Reference Angle for Cosine

For cosine in the second quadrant, it is negative and equal to the cosine of its reference angle. Therefore, \( \cos \frac{2\pi}{3} = -\cos \frac{\pi}{3} = -\frac{1}{2} \).

Step 4: Calculate Tangent from Sine and Cosine

The tangent of an angle is the ratio of sine to cosine, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). So, \( \tan \frac{2\pi}{3} = \frac{\sin \frac{2\pi}{3}}{\cos \frac{2\pi}{3}} = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = -\sqrt{3} \).

Key Concepts

Unit Circle

The unit circle is a powerful tool in trigonometry. It helps us visualize and calculate the values of trigonometric functions at various angles. Imagine a circle with a radius of 1, centered at the origin of a coordinate plane. This circle has special properties that make calculations straightforward.
In this circle, any angle can be represented as a point on the circle. The x-coordinate of this point gives the cosine of the angle, and the y-coordinate provides the sine. Since the radius is always 1, these coordinates are the exact values of the sine and cosine for the respective angle.

• Angles are typically measured in radians when using the unit circle.
• A full circle is equal to an angle of \(2\pi\) radians.
• Quadrants of the circle help determine the sign of the trigonometric functions.

The unit circle helps easily determine the positive or negative sign of trigonometric values depending on which quadrant the angle lies.

Reference Angles

Reference angles are angles created by drawing a line from the terminal side of an angle to the x-axis. This helps simplify the calculations of trigonometric functions, especially when the angle is not on the main axes.
When calculating functions for angles greater than \(\pi/2\), reference angles allow us to find equivalent angle measures in the first quadrant. The trigonometric function of any angle can be simplified using its reference angle, which always lies between 0 and \(\frac{\pi}{2}\).

• In the second quadrant, subtract the angle from \(\pi\) to get the reference angle.
• Reference angles retain the function's magnitude, while quadrant knowledge tells us the sign.
• This makes complex calculations easier by converting them into well-known angles.

By using reference angles, you simplify finding exact answers for trigonometric functions in other quadrants.

Sine

The sine function, \(\sin\), is one of the fundamental trigonometric functions. It relates to the y-coordinate of a point on the unit circle. It is particularly useful in calculating the height or opposite side in right angle triangles.
Sine values are positive in the first and second quadrants and are found using reference angles.

• For any angle \(\theta\) on the unit circle, \(\sin \theta\) is the y-coordinate.
• Sine reaches maximum at \(\frac{\pi}{2}\) with a value of 1, and minimum at \(-\frac{\pi}{2}\) with a value of -1.
• Periodic with a cycle of \(2\pi\).

At \(\frac{2\pi}{3}\), using the unit circle and reference angle \(\frac{\pi}{3}\), \(\sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2}\) because it's in the second quadrant.

Cosine

The cosine function, denoted \(\cos\), represents the x-coordinate of the unit circle. Cosine is key in calculating the hypotenuse or adjacent side in right triangles.
In the unit circle, cosine values are positive in the first and fourth quadrants and negative in the second and third. The reference angle for cosine helps determine these values efficiently.

• For any angle \(\theta\), \(\cos \theta\) is the x-coordinate.
• The function has maximum and minimum values of 1 and -1, respectively at \(0\) and \(\pi\).
• Cosine is also periodic with a cycle of \(2\pi\).

Smaller reference angles provide precise cosine values. For instance, \(\cos \frac{2\pi}{3} = -\frac{1}{2}\) using reference angle \(\frac{\pi}{3}\) in the second quadrant.

Tangent

Tangent, or \(\tan\), is the ratio of the sine to the cosine of an angle. It's valuable in determining slopes or heights and horizontal distances in right-angled contexts.
On the unit circle, the tangent function is unique because it doesn't directly correspond to a coordinate, but rather to their ratio.

• Tangent is positive in the first and third quadrants and negative in the second and fourth.
• Its values can vary drastically as cosine approaches zero.
• The function also repeats every \(\pi\) radians, half the cycle of sine and cosine.

For \(\frac{2\pi}{3}\), using \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), we find \(\tan \frac{2\pi}{3} = -\sqrt{3}\), due to the negative cosine in the second quadrant.