Problem 96 - A Graphical Approach to Precalculus with Limits

Question

For each expression, (a) write the function in terms of a function of the reference angle. (b) give the exact value, and (c) use a calculator to show that the decimal value or approximation for the given function is the same as the decimal value or approximation for your answer in part (b). $$\sin \frac{5 \pi}{3}$$

Step-by-Step Solution

Answer

\( \sin \frac{5\pi}{3} = -\frac{\sqrt{3}}{2} \).

Step 1: Identify the Reference Angle

The given angle is \( \frac{5\pi}{3} \). Since this angle is larger than \(\pi\), we find its reference angle by subtracting \(2\pi - \frac{5\pi}{3}\). Calculating gives us \(\frac{6\pi}{3} - \frac{5\pi}{3} = \frac{\pi}{3}\). Thus, the reference angle is \(\frac{\pi}{3}\).

Step 2: Determine the Sign of the Function

The given angle \(\frac{5\pi}{3}\) is in the fourth quadrant, where the sine function is negative. Therefore, \( \sin \frac{5\pi}{3} = -\sin \frac{\pi}{3} \).

Step 3: Find the Exact Value of the Sine Function

Using the reference angle \(\frac{\pi}{3}\), we know that \(\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\). Therefore, \(\sin \frac{5\pi}{3} = - \frac{\sqrt{3}}{2}\).

Step 4: Confirm Using a Calculator

Use a calculator to check that \(\sin \frac{5\pi}{3} \approx -0.866\) and the exact value \(-\frac{\sqrt{3}}{2} \approx -0.866\) match. This confirms the calculations.

Key Concepts

Reference Angle

Understanding the concept of a reference angle is crucial when dealing with trigonometric functions. A reference angle is the smallest angle between the terminal side of a given angle and the x-axis. It's always a positive acute angle, meaning it is between 0 and \(rac{\pi}{2}\) radians (or 0 and 90 degrees). Reference angles help us simplify trigonometric problems by allowing us to express an angle in terms of a more manageable acute angle.

To find the reference angle, you typically need to look at the relation of the given angle with the nearest x-axis. For example, if the given angle is negative or greater than \(rac{\pi}{2}\) radians, subtract or add enough multiples of \(\pi\) or \(2\pi\) to fit it within a complete rotation (0 to \(2\pi\)).

In the problem provided, the given angle is \(\frac{5\pi}{3}\), which is clearly more than one complete motion around the circle \(2\pi\). Therefore, subtracting \(2\pi - \frac{5\pi}{3}\) gives us \(\frac{\pi}{3}\), a direct and simple acute angle that functions well as a reference point.

Quadrants in Trigonometry

In trigonometry, the coordinate plane is divided into four quadrants, and each impacts the sign of trigonometric functions such as sine, cosine, and tangent. Understanding these quadrants helps identify the behavior of trigonometric functions across different angles.

• First Quadrant (0 to \(\frac{\pi}{2}\)): All trigonometric functions are positive.
• Second Quadrant (\(\frac{\pi}{2}\) to \(\pi\)): Only sine remains positive.
• Third Quadrant (\(\pi\) to \(\frac{3\pi}{2}\)): Tangent is positive.
• Fourth Quadrant (\(\frac{3\pi}{2}\) to \(2\pi\)): Cosine is positive.

Determining which quadrant an angle lies in helps decide the sign of the trigonometric expression. For \(\frac{5\pi}{3}\), it falls into the fourth quadrant, where sine is negative. This results in \ "\(\sin \frac{5\pi}{3} = -\sin \frac{\pi}{3}\)".

Knowing which functions are positive or negative in each quadrant aids in verifying solutions and understanding why certain signs appear in trigonometric equations.

Exact Value Calculation

Exact value calculations in trigonometry involve determining the exact numerical expression of a trigonometric function, rather than just approximating it with decimal numbers. This usually results in expressions involving square roots, fractions, or other constants such as \(\pi\).

The accuracy in these calculations ensures a precise understanding of trigonometric functions' values. Building a solid foundation with special angles, like \(0\), \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), and \(\frac{\pi}{2}\), is crucial, as their trigonometric values are often memorized due to their repetitiveness and exactitude.

For the angle \(\frac{\pi}{3}\), you should know that \(\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\). This value does not need estimation or approximation because it represents the exact value. Consequently, for \(\sin \frac{5\pi}{3}\), given it is in the fourth quadrant, the exact value becomes \ "\(-\frac{\sqrt{3}}{2}\)" ensuring all calculations maintain precision as verified with the calculator.