Problem 42 - Algebra 2

Question

Find the exact value of each trigonometric function. $$ \cos 30^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \( \cos 30^{\circ} \) is \( \frac{\sqrt{3}}{2} \).

Step 1: Recall Trigonometric Function Values

To find the exact value of \( \cos 30^{\circ} \), recall that it corresponds to one of the special angles in trigonometry. These special angles are 30°, 45°, and 60°, and their cosine values can be remembered using known trigonometric ratios.

Step 2: Identify the Special Angle Triangle

The angle 30° is part of a 30°-60°-90° triangle, which has side ratios of 1 (opposite to 30°), \( \sqrt{3} \) (adjacent to 30°), and 2 (hypotenuse).

Step 3: Calculate the Cosine of 30°

In a 30°-60°-90° triangle, the cosine of 30° can be calculated using the ratio: \( \cos 30^{\circ} = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2} \).

Step 4: Verify the Exact Value

Check the calculated exact value by ensuring it matches the known trigonometric table values for \( \cos 30^{\circ} \). The exact cosine value for 30° should be \( \frac{\sqrt{3}}{2} \), confirming the calculation is correct.

Key Concepts

Cosine Function

The cosine function is one of the foundational trigonometric functions, symbolized as \( \cos \). It relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. In simpler terms:

• The cosine of an angle in a right triangle is the length of the side next to the angle divided by the length of the longest side (the hypotenuse).
• It's part of the primary trio of trigonometric functions alongside sine (\( \sin \)) and tangent (\( \tan \)).

Cosine values can be positive or negative depending on the quadrant of the angle. However, for angles often encountered in 0° to 90°, the values will only be positive.
Commonly used in various fields, including physics and engineering, the cosine function is pivotal in analyzing wave patterns, rotational dynamics, and even in computer graphics. To remember cosine's identity, think of the formula: \[ \cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}} \] for a given angle \( \theta \).

30-60-90 Triangle

A 30-60-90 triangle is a specific type of right triangle with angles of 30 degrees, 60 degrees, and 90 degrees. This triangle is notable because its side lengths always follow a particular ratio: 1 : \( \sqrt{3} \) : 2. This makes it highly useful in trigonometry for quickly finding the sine, cosine, and tangent of 30° and 60° angles without needing a calculator.
Here's a quick breakdown of the side lengths:

• The shortest side, opposite the 30° angle, is 1.
• The side opposite the 60° angle has a length of \( \sqrt{3} \).
• The hypotenuse, opposite the 90° angle, is the longest, with a length of 2.

Understanding the 30-60-90 triangle's properties helps us derive the exact trigonometric values for these angles. This concept is central to geometry and often serves as a shortcut for solving more complex trigonometric problems.

Exact Values in Trigonometry

Exact values in trigonometry refer to the well-defined, non-decimal values of trigonometric functions for particular angles. Key angles, such as 30°, 45°, and 60°, have specific exact values that are frequently used and get memorized for ease.
For example:

• \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \sin 30^{\circ} = \frac{1}{2} \)
• \( \tan 30^{\circ} = \frac{1}{\sqrt{3}} \)

These values come from analyzing special triangles like the 30-60-90 triangle. Knowing the exact values allows students to quickly solve trigonometric problems without calculations.
They are vital in many applications, including calculus, physics, and computer science, whenever exact values are required instead of approximate decimal numbers.