Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-44,\) find the exact value. $$ \tan 30^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \( \tan 30^{\circ} \) is \( \frac{\sqrt{3}}{3} \).

Step 1: Recall the Definitions of Trigonometric Ratios

The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. For angle \( \theta \), this is written as \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \).

Step 2: Use the Tangent Value for Special Angles

The tangent value for commonly used angles like \( 30^{\circ}, 45^{\circ}, \) and \( 60^{\circ} \) are often memorized. The exact value of \( \tan 30^{\circ} \) is a known trigonometric identity.

Step 3: Remember Tangent of 30 Degrees

\( \tan 30^{\circ} \) is equal to \( \frac{1}{\sqrt{3}} \). This can also be rationalized to give \( \frac{\sqrt{3}}{3} \).

Step 4: Conclude with the Exact Value

Thus, the exact value of \( \tan 30^{\circ} \) is \( \frac{\sqrt{3}}{3} \).

Key Concepts

Tangent Function

The tangent function is one of the foundational trigonometric functions, often abbreviated as "tan." It relates an angle in a right triangle to the lengths of two of its sides. Specifically, for an angle \( \theta \), the tangent is defined as the ratio between the length of the side opposite the angle and the length of the side adjacent to it. This can be expressed mathematically as:

• \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)

Understanding this ratio is crucial when dealing with problems related to slopes, angles, and various geometry-related calculations. The tangent function is periodic, sharing this trait with other trigonometric functions, and has vertical asymptotes at certain points, producing a graph that repeats periodically every \(180^{\circ}\) (or \(\pi\) radians). The behavior of the tangent function can be visualized on the unit circle, where the tangent represents the length of the line segment from the circle to the tangent of the circle at a given angle.

Special Angles

Special angles in trigonometry are frequently used angles that have simple and easily remembered trigonometric values. These angles include \(0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}, \) and \(90^{\circ}\). Let's explore how they function:

• \(30^{\circ}:\) A commonly known triangle for this angle is a 30-60-90 triangle. The side ratios are 1: \(\sqrt{3}\): 2, which helps determine trig values.
• \(45^{\circ}:\) This angle forms an isosceles right triangle with side ratios of 1:1:\(\sqrt{2}\), often memorized for quick calculation.
• \(60^{\circ}:\) Similar to \(30^{\circ}\), a 30-60-90 triangle is used, just rotated to put \(60^{\circ}\) in focus for trig values.

These angles are particularly useful because their trigonometric values can often be found exactly, rather than approximated by decimals. This means they are seen frequently in a host of problems in trigonometry and calculus. Using known values of these angles also streamlines computations and assists in developing a deeper understanding of trigonometric functions.

Exact Values

In mathematics, exact values refer to values or expressions that are not rounded or approximated. These are often represented with fractions or square roots instead of decimal approximations.

• The tangent of \(30^{\circ}\) is a classic example. The exact value is \( \frac{1}{\sqrt{3}} \), which is often rationalized to give \( \frac{\sqrt{3}}{3} \).
• This concept is crucial in trigonometry, often providing cleaner solutions and preventing errors from numerical approximations.

Exact values are particularly relevant in higher mathematics fields such as calculus and complex analysis. They form the backbone of many math problems, simplifying expressions into a form that can be easily manipulated or analyzed further. Knowing the exact values for the tangent or other trigonometric functions validates solutions with high precision and accuracy. Therefore, mastering these exact values is a key step for any student learning trigonometry.