Problem 23 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

\( \frac{1 + \sqrt{3}}{2} \)

Step 1: Identify Trigonometric Values

To solve the expression \( \sin \frac{\pi}{6} + \cos \frac{\pi}{6} \), we need to know the values of \( \sin \frac{\pi}{6} \) and \( \cos \frac{\pi}{6} \). These are special angles on the unit circle.

Step 2: Evaluate \( \sin \frac{\pi}{6} \)

The value of \( \sin \frac{\pi}{6} \) is \( \frac{1}{2} \). This is a commonly memorized value from the unit circle for the angle \( 30^\circ \) or \( \frac{\pi}{6} \) radians.

Step 3: Evaluate \( \cos \frac{\pi}{6} \)

The value of \( \cos \frac{\pi}{6} \) is \( \frac{\sqrt{3}}{2} \). This is another standard value from the unit circle for the same angle.

Step 4: Add the Values

Now, add the values from Step 2 and Step 3: \( \sin \frac{\pi}{6} + \cos \frac{\pi}{6} = \frac{1}{2} + \frac{\sqrt{3}}{2} \).

Step 5: Simplify the Expression

The expression \( \frac{1}{2} + \frac{\sqrt{3}}{2} \) can be combined because they have a common denominator: \( \frac{1 + \sqrt{3}}{2} \). This is the simplest form of the expression.

Key Concepts

Understanding the Unit Circle

The unit circle is a fundamental concept in trigonometry. It helps us understand trigonometric functions such as sine and cosine. Imagine a circle with a radius of 1 unit centered at the origin of a graph. This is what we refer to as the unit circle.
You might wonder why the radius is set as 1 unit. This choice is deliberate because it makes calculations simpler and more intuitive.
Each point on the unit circle corresponds to an angle measured from the positive x-axis. The coordinates of these points are \( (\cos(\theta), \sin(\theta)) \), where \(\theta\) is the angle in radians.

• The x-coordinate represents the cosine of the angle \(\theta\).
• The y-coordinate represents the sine of the angle \(\theta\).

This simple idea allows us to map angles directly to their sine and cosine values, helping us quickly determine these values for special angles.

Special Angles on the Unit Circle

Special angles are key angles often used in trigonometry that have known sine and cosine values. These include angles like \(0, \pi/6, \pi/4, \pi/3,\) and \(\pi/2\), among others. These angles are called "special" because their trigonometric values are easy to memorize and commonly appear on the unit circle.
For instance:

• \(\frac{\pi}{6}\) (or 30°) is an angle whose sine is \(\frac{1}{2}\) and cosine is \(\frac{\sqrt{3}}{2}\).
• \(\frac{\pi}{4}\) (or 45°) results in both sine and cosine being \(\frac{\sqrt{2}}{2}\).
• \(\frac{\pi}{3}\) (or 60°) swaps the values of \(\frac{\pi}{6}\): sine is \(\frac{\sqrt{3}}{2}\) and cosine is \(\frac{1}{2}\).

By knowing these angles, you can solve various trigonometric problems without using a calculator. This knowledge plays a massive role when working on exercises like the one we have.

Sine and Cosine Values Simplified

Sine and cosine are two of the most important functions in trigonometry. They relate the angles of triangles to the lengths of their sides and extend to various applications including wave functions, sound, and even alternating current circuits.
Let's look closer at the sine and cosine for our special angle \(\frac{\pi}{6}\):

• Sine of an angle: The vertical coordinate on the unit circle, or the opposite side over the hypotenuse in a triangle. For \(\theta = \frac{\pi}{6}\), \(\sin \frac{\pi}{6} = \frac{1}{2}\).
• Cosine of an angle: The horizontal coordinate on the unit circle, or the adjacent side over the hypotenuse. For \(\theta = \frac{\pi}{6}\), \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\).

These values are crucial because they allow us to compute expressions effortlessly, as seen in the exercise.

Angle Conversion and its Importance

Angles can be measured in degrees or radians, and converting between these two is essential for solving trigonometric problems.
In trigonometry, it is often more convenient to use radians, especially as most mathematical formulas assume angles are in radians.

• Conversion rule: To convert from degrees to radians, multiply by \(\frac{\pi}{180}\).
• To convert from radians to degrees, multiply by \(\frac{180}{\pi}\).

Let's take the angle \(\frac{\pi}{6}\). If you want to find its degree counterpart, multiply it by \(\frac{180}{\pi}\) to get 30°. This equivalence ensures you can switch contexts as needed, making it easier to apply trigonometric principles across different problems. Recognizing that \(\pi/6\) radians is the same as 30° is highly valuable when aligning your methods across different exercises.