Problem 87 - Algebra and Trigonometry

Question

find the exact value of each expression. Write the answer as a single fraction. Do not use a calculator. $$ \sin \frac{\pi}{3} \cos \pi-\cos \frac{\pi}{3} \sin \frac{3 \pi}{2} $$

Step-by-Step Solution

Answer

The simplified result of the expression is \( \frac{-\sqrt3}{2} + \frac{1}{2} \).

Step 1: Recognize the Values

Start by recognizing the sine and cosine of specific angles. The values are as follows: \( \sin \frac{\pi}{3} = \frac{\sqrt3}{2} \), \( \cos \pi = -1 \), \( \cos \frac{\pi}{3} = \frac{1}{2} \), and \( \sin \frac{3 \pi}{2} = -1 \).

Step 2: Substitute Values

Substitute the known sine and cosine values into the original expression. The equation now looks like this: \( \frac{\sqrt3}{2} \cdot -1 - \frac{1}{2} \cdot -1 \).

Step 3: Simplify the Expression

Finally, simplify the expression. Multiply the numbers together: \( \frac{-\sqrt3}{2} + \frac{1}{2} \). The expression can't be reduced further because the terms are not like terms.

Key Concepts

Sine and Cosine Values

Trigonometric functions, such as sine and cosine, are fundamental in understanding and solving mathematical problems involving angles. These functions are based on the ratios of sides in a right triangle or the characteristics of circular motion. The sine of an angle corresponds to the vertical coordinate on the unit circle, whereas the cosine corresponds to the horizontal coordinate. Here are some standard values that are widely used:

• The sine of \( \frac{\pi}{3} \) is \( \frac{\sqrt{3}}{2} \), a value derived from an equilateral triangle, or from the corresponding point on the unit circle.
• The cosine of \( \frac{\pi}{3} \) is \( \frac{1}{2} \), also due to the same geometrical properties and unit circle location.
• The cosine of \( \pi \), a straight angle, is -1. This holds because a 180-degree rotation on the unit circle points directly left, opposite its origin point.
• Finally, the sine of \( \frac{3\pi}{2} \) is -1, as it's the bottom-most point on the unit circle, completing three-quarters of a full circle.

Angle Measures

Understanding angle measures in both degrees and radians is crucial for solving trigonometric expressions. Angles can be represented in these two units, with radians being more prevalent in calculus and higher mathematics.

• One complete revolution around a circle equals \( 2\pi \) radians or 360 degrees. Therefore, simpler angles like \( \frac{\pi}{3} \), \( \pi \), and \( \frac{3\pi}{2} \) are common in trigonometry.
• \( \frac{\pi}{3} \) radians is equivalent to 60 degrees, \( \pi \) radians to 180 degrees, perfectly splitting a circle in half.
• Similarly, \( \frac{3\pi}{2} \) radians, which translates to 270 degrees, is three-quarters around the circle.

Knowing these conversions helps in understanding and simplifying trigonometric problems.

Exact Trigonometric Values

Exact trigonometric values are fundamental in trigonometry as they allow for precise calculations without the use of approximations or calculators. These values are derived from geometric properties and symmetry of the unit circle:

• For example, \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \) and \( \cos \frac{\pi}{3} = \frac{1}{2} \) are exact values.
• Similarly, \( \cos \pi = -1 \) and \( \sin \frac{3\pi}{2} = -1 \) are determined by the specific angles and their positions on the unit circle.

Being able to recognize and substitute these exact values is critical in solving expressions like the one given, \( \sin \frac{\pi}{3} \cos \pi - \cos \frac{\pi}{3} \sin \frac{3\pi}{2} \). By replacing these with known fraction values, students can simplify calculations efficiently. In this case, those values help reduce the problem to a simple arithmetic, resulting in \( \frac{-\sqrt{3}}{2} + \frac{1}{2} \).