Problem 22 - Algebra 2 and Trigonometry

Question

In \(3-44,\) find the exact value. $$ \sin 180^{\circ} $$

Step-by-Step Solution

Answer

The exact value is 0.

Step 1: Understanding the Sine Function

The sine function, denoted by \( \sin \), is a trigonometric function that relates to the angle in a right triangle or on the unit circle. For angles measured in degrees, the sine function evaluates the y-coordinate of a point on the unit circle where the terminal side of the angle intersects the circle.

Step 2: Locate the Angle on the Unit Circle

The angle given is \(180^{\circ}\). On the unit circle, \(180^{\circ}\) corresponds to the point directly across from \(0^{\circ}\), along the negative x-axis.

Step 3: Identify the Coordinates on the Unit Circle

The unit circle has an equation \(x^2 + y^2 = 1\). For \(180^{\circ}\), the coordinates are \((-1, 0)\) because it lies on the negative x-axis.

Step 4: Determine the Sine Value

The sine of an angle is the y-coordinate of the corresponding point on the unit circle. Thus, for the angle \(180^{\circ}\), the sine value is the y-coordinate, which is \(0\).

Key Concepts

unit circle

The unit circle is a fundamental concept in trigonometry and is invaluable for understanding trigonometric functions. Imagine a circle with a radius of 1, centered at the origin of a coordinate plane. This circle is known as the unit circle. Since the radius is always 1, any point on the circle satisfies the equation \( x^2 + y^2 = 1 \). This property allows us to easily derive the values of trigonometric functions based on the positions of angles.Points on the unit circle are expressed as \((x, y)\), where each angle corresponds to a unique point. Using the unit circle, trigonometric functions like sine and cosine become the y and x coordinates of these points, respectively. This is why the unit circle is a powerful tool for evaluating trigonometric functions.

sine function

The sine function, often written as \( \sin \), connects angles to the y-coordinate of points on the unit circle. This is particularly useful when working with angles in right triangles or on the unit circle itself. One can visualize this by remembering that as an angle increases, the terminal side rotates and intersects the unit circle at a new point, whose y-coordinate gives us the sine value.Let's consider the angle \( 180^{\circ} \). On the unit circle, this angle is located directly on the negative x-axis, which corresponds to the point \((-1, 0)\). As the sine function is associated with the y-coordinate, for \(180^{\circ}\), \( \sin 180^{\circ} = 0 \).

• The sine function creates a wave pattern when plotted, repeating its values as the angle increases and decreases.
• It is crucial for modeling periodic phenomena, like sound waves or circular motion.

angle measurement

Angles are fundamental in trigonometry and can be measured in degrees or radians. Degrees are common in various aspects of math and everyday applications. One full rotation around a circle is \( 360^{\circ} \).To understand angle position, it's essential to know that:\

• \( 0^{\circ} \) is on the positive x-axis.
• \( 90^{\circ} \) is at the top of the circle, on the positive y-axis.
• \( 180^{\circ} \) is directly opposite \( 0^{\circ} \) on the negative x-axis.
• \( 270^{\circ} \) is at the bottom of the circle, on the negative y-axis.

These positions are pivotal for precisely finding the values of trigonometric functions, which depend on the coordinate positions of these angles on the unit circle. For instance, knowing that \( 180^{\circ} \) is at \((-1, 0)\) helps quickly determine the sine, cosine, and tangent values of this angle.