Problem 64 - Precalculus Essentials

Question

Describe the triangle used to find the trigonometric functions of \(30^{\circ}\) and \(60^{\circ}\).

Step-by-Step Solution

Answer

In the 30-60-90 triangle, the side lengths establish the trigonometric functions. For a \(30^{\circ}\) angle, sin is \(1 / 2\), cos is \(\sqrt{3} / 2\), and tan is \(1 / \sqrt{3}\). For a \(60^{\circ}\) angle, sin is \(\sqrt{3} / 2\), cos is \(1 / 2\), and tan is \(\sqrt{3}\).

Step 1: Draw the Right Triangle

Draw a right-angled triangle. Label the angles as \(30^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\). The \(30^{\circ}\) angle is opposite the shortest side, the \(60^{\circ}\) is opposite the side of length \(\sqrt{3}\), and the \(90^{\circ}\) angle is opposite the longest side (hypotenuse) of 2.

Step 2: Establish the Trigonometric Functions

The sine of an angle in a right triangle is the length of the opposite side divided by the length of the hypotenuse. The cosine of an angle is the length of the adjacent side divided by the length of the hypotenuse. The tangent of an angle is the length of the opposite side divided by the length of the adjacent side.

Step 3: Determine the Trigonometric Values

For the \(30^{\circ}\) angle, the sine is \(1 / 2\), the cosine is \(\sqrt{3} / 2\), and the tangent is \(1 / \sqrt{3}\). For the \(60^{\circ}\) angle, the sine is \(\sqrt{3} / 2\), the cosine is \(1 / 2\), and the tangent is \(\sqrt{3}\).

Key Concepts

Right Triangle

A right triangle is a special type of triangle that consists of three sides and three angles, with one angle being exactly 90 degrees. This 90-degree angle defines the right triangle, distinguishing it from other types of triangles. It forms when two lines meet perpendicular to each other. The sides of a right triangle include:

• Hypotenuse: The longest side, opposite the right angle.
• Opposite side: The side across from the angle we are focusing on.
• Adjacent side: The side that forms the angle along with the hypotenuse.

In our context, consider a right triangle with angles 30°, 60°, and 90°. Here, the hypotenuse is the longest side, often given a length based on a known ratio. Understanding the sides of a right triangle helps in establishing efficient calculations for trigonometric functions.

Angles of a Triangle

In any triangle, the sum of the angles is always 180 degrees. In the case of a right triangle, one of the angles is 90 degrees. This leaves the remaining two angles to sum up to 90 degrees.

• In a 30°-60°-90° triangle, aside from the right angle of 90°:
• The 30° angle is typically the smallest angle.
• The 60° angle is larger than the 30° angle but smaller than the right angle.

Understanding angles of a triangle is crucial in trigonometry because it directly influences the relationships between the sides of the triangle, which is the foundation for solving many trigonometric problems. Knowing specific angle measures, such as 30° and 60°, allows utilization of unique right triangle ratios helpful in determining trigonometric functions.

Sine and Cosine Calculations

Sine and cosine are fundamental trigonometric functions used to relate angles to side lengths in right triangles. These functions are based on ratios of different sides of a right triangle with respect to a specified angle.

Sine Function

The sine function calculates the ratio of the length of the opposite side to the hypotenuse. For an angle θ, it is written as:\( \sin(\theta) = \frac{\text{Opposite side}}{\text{Hypotenuse}} \)

Cosine Function

The cosine function calculates the ratio of the length of the adjacent side to the hypotenuse. It is expressed as:\( \cos(\theta) = \frac{\text{Adjacent side}}{\text{Hypotenuse}} \)

Examples with 30° and 60°

• For a 30° angle in our right triangle:
  • \( \sin(30^\circ) = \frac{1}{2} \)
  • \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \)
• For a 60° angle:
  • \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \)
  • \( \cos(60^\circ) = \frac{1}{2} \)

Recognizing these sine and cosine values for 30° and 60° angles is invaluable for quickly solving trigonometric problems without needing a calculator.