Problem 44 - Precalculus with Calculus Previews

Question

Find the given trigonometric function value. Do not use a calculator. $$ \sin \left(-180^{\circ}\right) $$

Step-by-Step Solution

Answer

\(\sin(-180^{\circ}) = 0\) because \(\sin(180^{\circ}) = 0\).

Step 1: Understanding Trigonometric Functions with Negative Angles

The sine function is an odd function, which means that \(\sin(-\theta) = -\sin(\theta).\)In this case, we need to find \(\sin(-180^{\circ})\) unusing the property of initial sine function.

Step 2: Determining \( \sin(180^{\circ}) \)

The angle \(180^{\circ}\) is on the negative x-axis. For sine, we are concerned with the y-coordinate of the point on the unit circle. At \(180^{\circ}\), the point is (-1, 0). Thus, \(\sin(180^{\circ}) = 0.\)

Step 3: Finding \( \sin(-180^{\circ}) \) Using the Property of Odd Functions

Using the fact that sine is an odd function, \(\sin(-180^{\circ}) = -\sin(180^{\circ})\). Since\(\sin(180^{\circ}) = 0,\) we have\(\sin(-180^{\circ}) = -0 = 0.\)

Key Concepts

Sine Function

The sine function is one of the most fundamental trigonometric functions in mathematics. It is defined for all angles and gives the y-coordinate of a point on the unit circle. For any angle \( \theta \), the sine function is written as \( \sin(\theta) \). The value of \( \sin(\theta) \) varies between -1 and 1, depending on the position of \( \theta \) around the unit circle.

In practical terms, the sine value tells you how far up or down the point is from the x-axis. If the sine value is positive, the point is above the x-axis. If it is negative, the point is below the x-axis. Understanding this can help visualize the behavior of trigonometric cycles, such as waves and oscillations in physics and engineering.

Unit Circle

The unit circle is a key concept in trigonometry and helps in understanding how trigonometric functions like sine and cosine work. It is a circle with a radius of 1 centered at the origin (0,0) of the coordinate plane.

The unit circle allows us to define angles and their trigonometric values in a straightforward manner. Each point on the unit circle corresponds to an angle \( \theta \), measured from the positive x-axis. The x-coordinate of this point gives the cosine of the angle, while the y-coordinate gives the sine.

At \( 180^{\circ} \), the point on the unit circle is (-1, 0). This means that \( \sin(180^{\circ}) = 0 \), as the y-coordinate here is 0. The unit circle is useful for visualizing and computing the values of sine and cosine for various angles without needing a calculator.

Odd Function

An odd function is a type of mathematical function that has a specific symmetry. The defining property of odd functions is that \( f(-x) = -f(x) \) for all x in the domain of the function. This symmetry means that if you were to graph the function, it would have rotational symmetry about the origin. In simpler words, flipping both x and y values gives the same result as flipping y values alone.

The sine function is a classic example of an odd function. This property is particularly useful when dealing with negative angles in trigonometry. Using the odd function property, we can determine that \( \sin(-\theta) = -\sin(\theta) \).

In our context, it means \( \sin(-180^{\circ}) = -\sin(180^{\circ}) \). Since \( \sin(180^{\circ}) = 0 \), we get \( \sin(-180^{\circ}) = 0 \) as well. This further illustrates how understanding properties like odd functions can simplify computations in trigonometry.