Problem 3 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding the Problem

We are asked to find the exact value of the cosine of 30 degrees. This involves using trigonometric identities or properties of special angles.

Step 2: Identifying Special Triangles

30 degrees is a common angle found in a 30-60-90 triangle. In such a triangle, the side opposite the 30-degree angle is half as long as the hypotenuse, and the side opposite the 60-degree angle has a length equal to the hypotenuse multiplied by \( \sqrt{3}/2 \).

Step 3: Using the Cosine Definition

In a right triangle, the cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse. For a 30-degree angle in a unit circle or a scaled 30-60-90 triangle, the adjacent side (next to the 30-degree angle and between it and the 60-degree angle) is \( \sqrt{3}/2 \) and the hypotenuse is 1.

Step 4: Calculating the Cosine

Using the ratio defined for cosine: \( \cos 30^{\circ} = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\sqrt{3}/2}{1} = \sqrt{3}/2 \).

Key Concepts

Trigonometric identities

Trigonometric identities are fundamental relationships between the trigonometric functions such as sine, cosine, and tangent which often make solving problems involving angles much easier. These identities serve as essential tools in trigonometry, allowing us to simplify complex expressions and solve equations. For example, one crucial identity is the Pythagorean identity, \( \sin^2\theta + \cos^2\theta = 1 \). This is applicable to any angle \(\theta\).
Another important identity is the cosine of complementary angles, which states \( \cos(90^\circ - \theta) = \sin \theta \). These identities not only help in understanding trigonometric concepts but also provide strategies for finding unknown angle measures and side lengths in various geometrical problems. Understanding these identities is the key to mastering trigonometry.

Special triangles

Special triangles refer to those right triangles that have specific angles which form easily recognizable side ratios. The two most common types of special triangles are the 45-45-90 triangle and the 30-60-90 triangle. Each has specific side lengths in relation to their angles, making calculations more straightforward.

• 45-45-90 Triangle: Each angle measures 45 degrees, and the sides opposite these angles are equal. If the legs of the triangle are both \( 1 \), the hypotenuse will be \( \sqrt{2} \).
• 30-60-90 Triangle: This includes angles measuring 30 degrees, 60 degrees, and 90 degrees. The ratios of their sides follow the pattern: 1 (opposite the 30-degree angle), \( \sqrt{3} \) (opposite the 60-degree angle), and 2 (the hypotenuse).

These properties of special triangles simplify many trigonometric computations, as illustrated in finding \( \cos 30^{\circ} \). In practice, special triangles offer a neat, quick reference for exact trigonometric values.

30-60-90 triangle

The 30-60-90 triangle is a type of special triangle that appears frequently in mathematics, due to its defined geometric properties and perfect side ratios. Knowing these can help solve a range of trigonometric problems.
In a 30-60-90 triangle:

• The side opposite the 30-degree angle is the shortest and is half the length of the hypotenuse.
• The side opposite the 60-degree angle is \( \sqrt{3}/2 \) times the hypotenuse.
• The hypotenuse itself is the longest side in the triangle.

These properties are grounded in geometry and are instrumental in understanding and deriving trigonometric measures like \( \cos 30^{\circ} \). Knowing that the length of the side adjacent to the 30-degree angle is \( \sqrt{3}/2 \) when the hypotenuse is 1 enables us to quickly find the cosine value.

Unit circle

The unit circle is a fundamental concept in trigonometry, representing a circle with a radius of 1 centered at the origin of a coordinate plane. It's a crucial visual tool for understanding how trigonometric functions such as sine, cosine, and tangent relate to angles.

• The circle’s circumference is tied to the angle measures in both degrees and radians.
• The x-coordinate of a point on the unit circle corresponds to the cosine of the angle formed with the positive x-axis.
• The y-coordinate corresponds to the sine of the angle.

For example, the point where the circle intersects the x-axis at a 30-degree angle from the origin gives us the cosine value. In this case, at 30 degrees (or \( \pi/6 \) radians), the point is \(( \sqrt{3}/2, 1/2)\), showing that \( \cos 30^{\circ} = \sqrt{3}/2 \). This colocation of points and angles makes the unit circle a vital tool in teaching, visualizing, and solving trigonometric equations efficiently.