Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-12\) , find the exact function value of each of the following if the measure of the angle is given in radians. $$ \tan \frac{\pi}{6} $$

Step-by-Step Solution

Answer

The exact function value of \(\tan \frac{\pi}{6}\) is \(\frac{\sqrt{3}}{3}\).

Step 1: Understanding the problem

We need to find the exact value of the tangent function at an angle of \(\frac{\pi}{6}\) radians. This involves using known values from the unit circle or trigonometric identities.

Step 2: Recall the Unit Circle Values

Familiarize yourself with the unit circle values where \(\frac{\pi}{6}\) radians is equivalent to 30 degrees. From the unit circle, we know that for 30 degrees, the sine is \(\frac{1}{2}\) and the cosine is \(\frac{\sqrt{3}}{2}\).

Step 3: Use the Tangent Identity

The tangent of an angle is defined as the sine divided by the cosine. Therefore, \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) for any angle \(\theta\).

Step 4: Apply the Identity to the Given Angle

With \(\theta = \frac{\pi}{6}\), substitute the sine and cosine values: \(\tan \frac{\pi}{6} = \frac{\sin \frac{\pi}{6}}{\cos \frac{\pi}{6}} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}\).

Step 5: Simplify the Expression

Simplify the expression by multiplying the numerator and the denominator by \(2\) to eliminate the fractions: \(\frac{1}{2} \div \frac{\sqrt{3}}{2} = \frac{1 \times 2}{\sqrt{3} \times 2} = \frac{1}{\sqrt{3}}\).

Step 6: Rationalize the Denominator

Multiply by \(\frac{\sqrt{3}}{\sqrt{3}}\) to rationalize the denominator: \(\frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}\).

Key Concepts

Unit Circle

The unit circle is a powerful tool in trigonometry that helps to find the values of trigonometric functions for common angles. Imagine a circle with a radius of 1 centered at the origin of a coordinate plane.
This circle is called the unit circle. It helps us determine the coordinates of points on the circle, where each point corresponds to an angle measured from the positive x-axis.
For example, on the unit circle, the angle

• \(rac{\pi}{6}\) radians (or 30 degrees) corresponds to the point \((\frac{\sqrt{3}}{2}, \frac{1}{2})\).

These coordinates are essential because:

• The x-coordinate represents the cosine of the angle.
• The y-coordinate represents the sine of the angle.

The simplicity of the unit circle allows us to quickly recall trigonometric values without having to calculate them repeatedly. It is a staple for solving problems involving radians and degrees alike.

Tangent Function

The tangent function is one of the primary trigonometric functions used in mathematics. It is defined using the sine and cosine functions:\[\tan \theta = \frac{\sin \theta}{\cos \theta}\]This function expresses the ratio of the sine of an angle to the cosine of the same angle.
In our original exercise, we applied this definition to find \( \tan \frac{\pi}{6} \).

• The sine of \( \frac{\pi}{6} \) is \( \frac{1}{2} \).
• The cosine of \( \frac{\pi}{6} \) is \( \frac{\sqrt{3}}{2} \).

By dividing these values, we obtained:\[\tan \frac{\pi}{6} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}}\]And after rationalizing the denominator:\[\tan \frac{\pi}{6} = \frac{\sqrt{3}}{3}\]Tangent is periodic, meaning it repeats its values every \( \pi \) radians, and is undefined where the cosine is zero.

Angle in Radians

Radians are another way to measure angles, commonly used in calculus and trigonometry due to their relationship with the unit circle.

• One full revolution around a circle equals \(2\pi\) radians.
• A half circle, or straight line, equals \(\pi\) radians.
• A right angle, or \(90\) degrees, equals \(\frac{\pi}{2}\) radians.

In our problem, \(\frac{\pi}{6}\) radians is equivalent to \(30\) degrees. Radians are beneficial because they make the mathematical relationship between the angle and arc length in a circle more direct.
One radian is the angle made when the arc length is equal to the radius of the circle. This makes calculations involving calculus simpler and more natural when using radians instead of degrees.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variables involved, offering relationships between trigonometric functions.
These identities help simplify complex trigonometric expressions and solve equations.

• Basic identities include \( \sin^2 \theta + \cos^2 \theta = 1 \), which is derived from the Pythagorean Theorem.
• Another common identity is \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), used in the original exercise to evaluate \( \tan \frac{\pi}{6} \).
• Reciprocal identities include \( \csc \theta = \frac{1}{\sin \theta} \), \( \sec \theta = \frac{1}{\cos \theta} \), and \( \cot \theta = \frac{1}{\tan \theta} \).

Understanding these identities is crucial, as they are frequently used in trigonometry, calculus, and various fields of engineering and physics to solve problems involving angles, circles, and oscillatory motion.
These identities provide the foundation for more complex and advanced trigonometric problems.