Question

Use the \(30^{\circ}-60^{\circ}-90^{\circ}\) and \(45^{\circ}-45^{\circ}-90^{\circ}\) triangles to find each value. Round to four decimal places, if necessary. \(\cos 60^{\circ}\)

Short answer

\(\cos 60^{\circ} = \frac{1}{2}\).

Step-by-step solution

Understanding the Triangle

The problem asks us to use a special triangle, specifically the \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, to find \(\cos 60^{\circ}\). In these triangles, the sides have a specific ratio: the side opposite the \(30^{\circ}\) angle is 1, the side opposite the \(60^{\circ}\) angle is \(\sqrt{3}\), and the hypotenuse is 2.

Identify the Relevant Side

The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the hypotenuse. For \(\cos 60^{\circ}\), the side adjacent to \(60^{\circ}\) is 1, and the hypotenuse is 2.

Calculate Cosine

Apply the cosine definition: \(\cos 60^{\circ} = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{1}{2}\).

Round the Value

No rounding is necessary since \(\frac{1}{2}\) is already in simple form.

Key concepts

30-60-90 triangle

The 30-60-90 triangle is a special type of right triangle with unique properties that make calculations easier. It has angles of 30 degrees, 60 degrees, and 90 degrees.

The side lengths in a 30-60-90 triangle are always in the ratio of 1: \(\sqrt{3}\): 2. This means:

• The shortest side, opposite the 30-degree angle, is 1 unit.
• The longer leg, opposite the 60-degree angle, measures \(\sqrt{3}\) units.
• The hypotenuse, opposite the 90-degree angle, is the longest side and is 2 units long.

This ratio is derived from the properties of the triangle and helps us easily calculate the values of trigonometric functions like the cosine, sine, and tangent for the angles of 30 and 60 degrees. Knowing this ratio can simplify complex problems involving right triangles with these angles.

cosine function

The cosine function is one of the basic trigonometric functions and is key to understanding relationships in right triangles.

For a given angle, the cosine function represents the ratio between the length of the adjacent side to the hypotenuse in a right-angled triangle. It is defined as:

• \(\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}\)

In a 30-60-90 triangle, knowing the side lengths, you can easily determine the cosine of 30 or 60 degrees. For example, \(\cos(60^{\circ})\) is calculated using the adjacent side (length of 1) and hypotenuse (length of 2), resulting in \(\frac{1}{2}\).

This calculation showcases how special triangles like the 30-60-90 triangle can simplify seemingly complex trigonometric calculations. This function is vital in various fields such as mathematics, physics, and engineering, making it a fundamental concept for students to understand.

trigonometric ratios

Trigonometric ratios are essential in learning how the sides and angles of a right triangle relate to each other. They provide a mathematical way to express these relationships:

• Sine function (sin): The ratio of the length of the opposite side to the hypotenuse. \(\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}\)
• Cosine function (cos): The ratio of the length of the adjacent side to the hypotenuse. \(\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}\)
• Tangent function (tan): The ratio of the length of the opposite side to the adjacent side. \(\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}\)

In a -angle triangle like a 30-60-90 triangle, these ratios are particularly convenient to use because the side lengths are known. For example, the sine of 30 degrees can be easily found as \(\frac{1}{2}\), while the tangent of 60 degrees is \(\sqrt{3}\). These ratios help solve many types of problems, from calculating distances to understanding waves, in both school and real-world applications.