Problem 22 - Algebra and Trigonometry

Question

9–32 Find the exact value of the trigonometric function. $$\sin \frac{5 \pi}{3}$$

Step-by-Step Solution

Answer

The exact value is \(-\frac{\sqrt{3}}{2}\).

Step 1: Identify the Angle's Reference

The angle given is \( \frac{5\pi}{3} \). Converting it to degrees, \( \frac{5\pi}{3} \times \frac{180}{\pi} = 300 \) degrees. Since 300 degrees is in the fourth quadrant, we will need to find its reference angle.

Step 2: Calculate the Reference Angle

The reference angle of 300 degrees is \( 360 - 300 = 60 \) degrees. In the fourth quadrant, only the cosine is positive, meaning the sine will be negative with the same magnitude as at 60 degrees.

Step 3: Use Known Trigonometric Values

From trigonometric values, \( \sin{60^\circ} = \frac{\sqrt{3}}{2} \). Since the angle \( \frac{5\pi}{3} \) is in the fourth quadrant, the sine value will be negative.

Step 4: Write the Exact Value

Thus, the exact value of \( \sin \frac{5\pi}{3} \) is \(-\frac{\sqrt{3}}{2}\).

Key Concepts

Understanding Reference Angles

Reference angles are crucial when dealing with trigonometric functions of any angle. A reference angle is the smallest angle that an angle makes with the x-axis, always between 0 and 90 degrees. It helps to simplify the calculation of trigonometric values for angles in different quadrants, by relating them to the more commonly known trigonometric values of the acute angles.
To find the reference angle, we need to consider the initial angle in standard position, meaning starting from the positive x-axis.

• If the angle is in the first quadrant, the angle itself is the reference angle.
• In the second quadrant, subtract the angle from 180 degrees.
• For angles in the third quadrant, subtract 180 degrees from the angle.
• And, for the fourth quadrant, subtract the angle from 360 degrees.

The procedure emphasizes subtracting the angle suitably based on which quadrant the angle lies. For example, a 300 degree angle lies in the fourth quadrant, making its reference angle 60 degrees.

Exploring Quadrants in Trigonometry

The study of quadrants helps in analyzing the signs and values of trigonometric ratios. The coordinate plane is divided into four quadrants which rotate counterclockwise starting from the positive x-axis. Each quadrant has specific characteristics which affect the signs of sine, cosine, and tangent functions.

• First Quadrant: All trigonometric functions (sine, cosine, tangent) are positive here.
• Second Quadrant: Sine is positive, while cosine and tangent are negative.
• Third Quadrant: Tangent is positive. Sine and cosine are negative.
• Fourth Quadrant: Cosine is positive, but sine and tangent are negative.

This knowledge is critical in finding trigonometric values of angles outside the 0 to 90 degrees range. In our example, when dealing with an angle in the fourth quadrant, the sine function will be negative, indicating how understanding quadrants helps to determine the sign of trigonometric functions.

The Basics of the Sine Function

The sine function is one of the primary trigonometric functions used to represent the y-coordinate of a point on the unit circle. It measures the ratio of the length of the opposite side to the hypotenuse in a right triangle. This makes it essential in various applications involving angles and circles.
The sine function has specific characteristics. It varies between -1 and 1 as angles transition from 0 to 360 degrees, periodically repeating its wave-like pattern. In different quadrants:

• In the first quadrant, sine values are positive.
• In the second quadrant, values remain positive.
• In the third quadrant, sine values become negative.
• In the fourth quadrant, sine values remain negative.

This knowledge helps in predicting the behavior of the sine function around the unit circle, making it easier to deduce its exact value in various quadrants.

Exact Values of Trigonometric Functions

Trigonometric functions of certain angles have exact values. These values are often computed for standard angles like 30, 45, 60, 90 degrees, and their variants in radians. They serve as benchmarks for calculating and understanding other angles.
For sine:

• At 0 degrees (or 0 radians), \(\sin{0} = 0\).
• At 30 degrees (or \(\frac{\pi}{6}\) radians), \(\sin{30^\circ} = \frac{1}{2}\).
• At 45 degrees (or \(\frac{\pi}{4}\) radians), \(\sin{45^\circ} = \frac{\sqrt{2}}{2}\).
• At 60 degrees (or \(\frac{\pi}{3}\) radians), \(\sin{60^\circ} = \frac{\sqrt{3}}{2}\).
• At 90 degrees (or \(\frac{\pi}{2}\) radians), \(\sin{90^\circ} = 1\).

Using these known values, one can derive the sine or cosine of other angles by considering the reference angle and quadrant placement, as with finding \(\sin{\frac{5\pi}{3}}\). Understanding these exact values enables quick and accurate evaluations of trigonometric expressions.