Problem 85 - Algebra and Trigonometry

Question

use reference angles to find the exact value of each expression. Do not use a calculator. $$ \sin \left(-\frac{17 \pi}{3}\right) $$

Step-by-Step Solution

Answer

The exact value of the expression is 0.

Step 1: Understand the Periodicity

The sine function has a periodicity of \(2\pi\), which means that \(\sin(x) = \sin(x + 2\pi*n)\) for any integer \(n\).

Step 2: Find the Equivalent Angle

By using the periodicity, determine an equivalent angle which lies in the interval \([-2\pi, 2\pi]\). Since the sine function is periodic every \(2\pi\), one can subtract \(2\pi*5\) from \(-\frac{17\pi}{3}\) to get this reference angle: \(-\frac{17\pi}{3} + 2\pi*5 = \pi\).

Step 3: Compute the Sine of Reference Angle

Lastly, evaluate the sine of the reference angle found, which is \(\sin(\pi)\).

Key Concepts

Periodicity of the Sine Function

Understanding the periodicity of the sine function is crucial when working with trigonometry problems. Periodicity refers to the characteristic of a function to repeat its values at regular intervals or periods. For the sine function, this period is precisely \(2\pi\), which means that the sine value repeats every \(2\pi\) radians. Mathematically, we express this as \( \sin(x) = \sin(x + 2\pi\cdot n) \) for any integer \(n\).

This property is incredibly useful because it allows us to simplify complex trigonometric expressions. If you're asked to find the sine of an angle outside the standard range of \([0, 2\pi]\), you can use periodicity to 'wrap around' and find an equivalent angle within the standard range. This equivalent angle will produce the same sine value as the original angle.
For instance, we can simplify \( \sin\left(-\frac{17\pi}{3}\right) \) by adding or subtracting multiples of \(2\pi\) until the angle falls within the standard range. This step is critical because it means you don't have to deal with potentially unfamiliar or unwieldy angles when calculating trigonometric values.

Finding Equivalent Angles

When solving trigonometric functions, finding equivalent angles that fall within the principal branch \([-\pi, \pi]\) or the standard range \([0, 2\pi]\) can greatly simplify calculations. This process involves adding or subtracting multiples of the function's period until the given angle is within the desired range.

In the exercise, we dealt with \( \sin\left(-\frac{17\pi}{3}\right) \) and we wanted to find an equivalent angle to make it simpler to handle. By recognizing that the sine function repeats every \(2\pi\), we subtracted \(2\pi\) multiple times from the given angle until it was within \([-2\pi, 2\pi]\). Specifically, subtracting \(2\pi\times5\) from \(-\frac{17\pi}{3}\) brought us to our reference angle of \(\pi\), which is within the principal branch.

Why Find Equivalent Angles?

Finding an equivalent angle is not just a mathematical trick; it's a practical approach to deal with angles of any size. By doing so, students are able to utilize the standard trigonometric values that are commonly memorized, making it easier to compute values mentally or apply further trigonometric identities.

Calculating Sine Without a Calculator

Calculating the sine of an angle without a calculator may seem daunting, but with the right knowledge, it’s completely manageable. A deep understanding of reference angles and the unit circle allows one to find sine values for commonly known angles manually.

A reference angle is the acute angle that a standard position angle makes with the x-axis. Common reference angles are \(0\), \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), and \(\frac{\pi}{2}\), and their corresponding sine values are easy to remember. Moreover, the sine function is symmetric about the y-axis, which means that \(\sin(-\theta) = -\sin(\theta)\), and it is periodic, thus \(\sin(\theta + 2\pi\cdot n) = \sin(\theta)\) for any integer \(n\).
By using the periodicity and symmetry properties, along with knowing sine values for a set of reference angles, you can quickly evaluate the sine function. In our exercise example, we reduced \(\sin\left(-\frac{17\pi}{3}\right)\) to \(\sin(\pi)\), a known reference angle for which the value of sine is zero. This is how one can calculate sine without a calculator—it’s all about strategically utilizing trigonometric properties and reference angles.