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. \(\tan 60^{\circ}\)

Short answer

\( \tan 60^{\circ} \approx 1.7321 \)

Step-by-step solution

Understanding the Problem

We need to find the value of \( \tan 60^{\circ} \). In trigonometry, \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \). We will use the properties of a special right triangle to find this.

Identify the Special Triangle

Since we need \( \tan 60^{\circ} \), we will use a \( 30^{\circ}-60^{\circ}-90^{\circ} \) triangle, where the sides are in a specific ratio: opposite \(60^{\circ}\) is \(\sqrt{3}\), opposite \(30^{\circ}\) is \(1\), and the hypotenuse is \(2\).

Calculate \( \tan 60^{\circ} \)

In the \( 30^{\circ}-60^{\circ}-90^{\circ} \) triangle, the side opposite \(60^{\circ}\) is \(\sqrt{3}\) and the side adjacent to \(60^{\circ}\) is \(1\). Therefore, \( \tan 60^{\circ} = \frac{\sqrt{3}}{1} = \sqrt{3} \).

Rounding the Result

Since \( \tan 60^{\circ} = \sqrt{3} \) and \( \sqrt{3} \approx 1.7321 \) when rounded to four decimal places, we have \( \tan 60^{\circ} \approx 1.7321 \).

Key concepts

Trigonometric Ratios

In trigonometry, understanding the relationship between the sides of a right triangle and its angles is crucial. These relationships are represented by trigonometric ratios. Specifically, for right triangles, we have three primary trigonometric ratios:

• Sine (sin): The ratio of the length of the side opposite the angle to the hypotenuse.
• Cosine (cos): The ratio of the length of the adjacent side to the hypotenuse.
• Tangent (tan): The ratio of the length of the opposite side to the adjacent side.

These ratios help in solving various problems related to angles and sides of triangles.
This makes them essential tools in mathematics and geometry. When working with specific angles within right triangles, like in special right triangles, these ratios become particularly useful.

30-60-90 Triangle

A 30-60-90 triangle is a special type of right triangle with angles measuring exactly 30 degrees, 60 degrees, and 90 degrees. The sides of a 30-60-90 triangle always follow a fixed ratio:

• The side opposite the 30° angle is the shortest and is often denoted as 1.
• The side opposite the 60° angle is longer, equal to \( \sqrt{3} \).
• The hypotenuse, opposite the 90° angle, is the longest side, equal to 2.

Understanding this ratio is essential because it allows us to find missing side lengths and angles quickly. It also simplifies solving trigonometric functions, such as finding the tangent of 60 degrees, due to the consistent relationship between the sides.

Tan 60 Degrees

The tangent function, often abbreviated as "tan," is one of the primary trigonometric ratios. To find \( \tan 60^{\circ} \), we focus on a 30-60-90 triangle. In this triangle:

• The side opposite the 60° angle is \( \sqrt{3} \).
• The side adjacent to the 60° angle is 1.

Thus, \( \tan 60^{\circ} \) can be calculated using the formula:\[ \tan 60^{\circ} = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sqrt{3}}{1} = \sqrt{3} \]When expressed as a decimal, \( \sqrt{3} \) is approximately 1.7321. Therefore, \( \tan 60^{\circ} \approx 1.7321 \).This calculation reinforces the usefulness of understanding the side ratios of special right triangles.