Problem 25 - Algebra and Trigonometry

Question

Evaluate the expression without using a calculator. $$ \sin \frac{\pi}{6}+\cos \frac{\pi}{6} $$

Step-by-Step Solution

Answer

The expression evaluates to \(\frac{1 + \sqrt{3}}{2}\).

Step 1: Identify Known Trigonometric Values

Recall that specific angles such as \( \frac{\pi}{6} \) have well-known trigonometric values. We know from the unit circle or trigonometric table that \( \sin \frac{\pi}{6} = \frac{1}{2} \) and \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \).

Step 2: Substitute Trigonometric Values into the Expression

Now that we have the values of \( \sin \frac{\pi}{6} \) and \( \cos \frac{\pi}{6} \), we substitute them into the expression: \[ \sin \frac{\pi}{6} + \cos \frac{\pi}{6} = \frac{1}{2} + \frac{\sqrt{3}}{2} \].

Step 3: Simplify the Expression

To simplify the expression, since both terms have a common denominator of 2, we can combine them into a single fraction:\[ \frac{1}{2} + \frac{\sqrt{3}}{2} = \frac{1 + \sqrt{3}}{2} \].

Key Concepts

Trigonometric Values

Understanding basic trigonometric values simplifies the evaluation of expressions involving sine and cosine. These trigonometric functions measure relationships in right-angled triangles or on the unit circle. Some angles, like \( \frac{\pi}{6} \), have easily recognizable values:

• \( \sin \frac{\pi}{6} = \frac{1}{2} \)
• \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \)

Knowing these values helps you solve problems quickly without referencing tables or calculators. These angles come up often, so memorize them for increased efficiency.

Unit Circle

The unit circle is a powerful tool in trigonometry that represents all possible values of sine, cosine, and tangent for each angle. The circle has a radius of 1, which makes it easy to derive the trigonometric values just by looking at coordinates of points along the circle. Layout:

• Angles are measured from the positive x-axis, moving counterclockwise.
• Coordinates (\( x, y \)) represent (\( \cos \theta, \sin \theta \)).

For \( \frac{\pi}{6} \), the coordinates are \( (\frac{\sqrt{3}}{2}, \frac{1}{2}) \). This shows the cosine and sine values directly, making it a quick reference for evaluating trigonometric expressions.

Simplifying Fractions

Simplifying fractions involves combining terms over a common denominator. This step is crucial when dealing with trigonometric expressions like \[ \frac{1}{2} + \frac{\sqrt{3}}{2} \]. Since both terms share a denominator, you can add the numerators directly:

• Combine numerators: \( 1 + \sqrt{3} \).
• Keep the common denominator: 2.

The result is \( \frac{1 + \sqrt{3}}{2} \). Simplifying helps in achieving a neat solution, presenting expressions in their simplest form for better understanding and further mathematical manipulation.