Problem 82 - Precalculus : Building Concepts and Connections

Question

Find the exact value of each expression without using a calculator. $$2 \sin \frac{\pi}{6}-\cos \frac{\pi}{3}$$

Step-by-Step Solution

Answer

The exact value of the expression \(2 \sin \frac{\pi}{6}-\cos \frac{\pi}{3}\) is \(\frac{1}{2}\)

Step 1: Substitute Trigonometric Values

First, replace \(\sin \frac{\pi}{6}\) and \(\cos \frac{\pi}{3}\) with their known standard value. The sin of \(\frac{\pi}{6}\) is \(\frac{1}{2}\) and the cos of \(\frac{\pi}{3}\) is \(\frac{1}{2}\). This transforms the expression to \(2 * \frac{1}{2} - \frac{1}{2}\)

Step 2: Perform Operations

Carry out the operations step by step. Starting with multiplication, \(2 * \frac{1}{2}\) results in \(1\). Then perform the subtraction, \(1-\frac{1}{2}\) results in \(\frac{1}{2}\)

Key Concepts

Trigonometric Values

Trigonometric values are numerical outcomes of trigonometric functions, such as sine (\textbackslash(sin)), cosine (\textbackslash(cos)), and tangent (\textbackslash(tan)), for specific angle measures. These functions are fundamental in understanding the relationships between the angles and sides of triangles, especially right-angled triangles.

Rather than relying on calculators, it's essential to become familiar with the exact values of these functions for common angles, such as \(0\), \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), and \(\frac{\pi}{2}\) radians, which correspond to 0°, 30°, 45°, 60°, and 90° degrees respectively. These exact values are derived from the geometric properties of triangles and the unit circle.

Unit Circle

The unit circle is a circle with a radius of one unit, centered at the origin of a coordinate plane. It is a powerful tool for understanding the trigonometric functions and their values because every point along the circumference of the unit circle corresponds to a pair of (cosine, sine) coordinates.

To find trigonometric values using the unit circle, imagine a line segment drawn from the origin to a point on the circle's edge, forming an angle with the positive x-axis. The x-coordinate of the point is the cosine value, and the y-coordinate is the sine value for the angle. Recognizing that specific angles yield simple rational numbers or square root values helps us memorize these coordinates and hence the sine and cosine values for those angles.

Basic Trigonometry

Basic trigonometry involves understanding the definitions of trigonometric functions and how to apply them to calculate unknown sides or angles in right-angled triangles. It also includes knowing how to manipulate these functions using algebraic identities and properties.

In the context of the given exercise, knowing that the sine of an angle in a right-angled triangle is the length of the opposite side over the hypotenuse, and the cosine is the adjacent side over the hypotenuse, forms the foundation for finding their values. Learning these basic trigonometric concepts and relationships is key for solving a variety of problems in mathematics and its applications in fields such as physics, engineering, and architecture.