Problem 41 - Precalculus: A Concise Course

Question

Evaluate the trigonometric function using its period as an aid. $$ \sin \left(-\frac{8 \pi}{3}\right) $$

Step-by-Step Solution

Answer

\(\sin(-8\pi / 3) = \sqrt{3}/2\)

Step 1: Identification of Periodic Function

The given function is a trigonometric function \(\sin(-8\pi / 3)\). We know that sine is a periodic function with a period of \(2\pi\). This means that if you go \(2\pi\) units in either direction on the x-axis, the y-value (or output) will be the same.

Step 2: Dividing the Angle by the Period

In the given function, the angle is \(-8\pi/3\). We can divide this value by the period of sine function i.e. \(2\pi\) to estimate the number of complete cycles the function passes through. Doing this gives \((-8\pi/3)/(2\pi)\) = -4/3 cycles.

Step 3: Calculate the Remaining Angle

We know from step 2 that there are exactly -4/3 cycles. But to return to an equivalent angle in the range from 0 to \(2\pi\), one full cycle (or \(2\pi\) radians) needs to be added. This gives us the equivalent positive angle as \(2\pi - 8\pi / 3\), which simplifies to \(2\pi/3\) radian.

Step 4: Evaluate the Function

Finally, we evaluate \(\sin(2\pi / 3)\). Recall that \(\sin\theta\) returns the y-coordinate of the point on the unit circle that is an angle theta counter-clockwise from the point (1,0), so we know \(\sin(2\pi / 3) = \sqrt{3}/2\).

Key Concepts

Periodic Functions

In mathematics, a **periodic function** is a function that repeats its values at regular intervals or periods. The concept is crucial when dealing with trigonometric functions such as sine and cosine. Periodic functions have a vast range of applications, from signal processing to the study of waves and vibrations.

Key Characteristics of Periodic Functions:

• Repetition: The function values repeat after a specific interval known as the period.
• Trigonometric Functions: Functions like sine and cosine are inherently periodic with a period of \(2\pi\).

The importance of periodicity in solving trigonometric problems is significant. It allows calculation of trigonometric values for any large angle by reducing it to an equivalent smaller angle within a singular cycle of the function.

Sine Function

The **sine function** is one of the primary functions in trigonometry, represented as \(\sin(\theta)\). It is used to find the y-coordinate on the unit circle for a given angle. The sine function is periodic and repeats its cycle every \(2\pi\) radians.

Characteristics of Sine Function:

• The range of \(\sin(\theta)\) is from -1 to 1.
• It starts at 0 when \(\theta = 0\), reaches a maximum of 1 at \(\pi/2\), returns to 0 at \(\pi\), and a minimum of -1 at \(3\pi/2\).
• The opposite values are taken at opposite angles, i.e., \(\sin(-\theta) = -\sin(\theta)\).

Understanding the sine function involves recognizing its graphical form—including its waves that oscillate upwards and downwards—and its symmetry. Knowing the symmetry properties helps in evaluating sine for negative angles, like in the exercise \(\sin\left(-\frac{8\pi}{3}\right)\).

Unit Circle

The **unit circle** is a circle with a radius of 1 centered at the origin of the coordinate plane. It is fundamental to trigonometry, serving as a visualization tool for the sine, cosine, and tangent functions.

Key Aspects of the Unit Circle:

• The angle \(\theta\) on the unit circle is measured from the positive x-axis, moving counter-clockwise.
• The x-coordinate corresponds to \(\cos(\theta)\) and the y-coordinate corresponds to \(\sin(\theta)\).
• Full rotation around the circle is \(2\pi\) radians or 360 degrees.

The unit circle assists in finding trigonometric function values by helping to locate points corresponding to particular angles. For \(2\pi/3\) radians, the sine value, or y-coordinate, can be directly found on the unit circle as \(\sqrt{3}/2\).

Radians

**Radians** are a unit of angular measure used in much of mathematics. Unlike degrees, radians provide a direct relationship between the radius of a circle and the arc length.

Benefits of Using Radians:

• A full circle is \(2\pi\) radians, which is approximately 6.283 radians.
• Radians allow for simpler and more natural formulas in calculus and other mathematics.
• Angular displacement can be easily related to arc length with the formula \(\text{Arc Length} = \text{Radius} \times \text{Angle in Radians}\).

Understanding radians is essential for evaluating trigonometric functions with angles given in radians, such as \(-8\pi/3\). This conversion helps simplify the angle to its equivalent position within one rotation of the unit circle, aiding in easy calculation of trigonometric values.