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

Short answer

\( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \).

Step-by-step solution

Identify the Special Triangle

To find \( \sin 60^{\circ} \), we use the properties of the \( 30^{\circ}-60^{\circ}-90^{\circ} \) triangle. In this triangle, the side opposite \( 30^{\circ} \) is \( x \), the side opposite \( 60^{\circ} \) is \( x\sqrt{3} \), and the hypotenuse is \( 2x \).

Use the Sine Definition

The sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. For \( \sin 60^{\circ} \), this is \( \sin 60^{\circ} = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{x\sqrt{3}}{2x} \).

Simplify the Ratio

Simplify the ratio \( \frac{x\sqrt{3}}{2x} \). The \( x \) in the numerator and denominator cancels out, leaving \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \).

Key concepts

Trigonometric Ratios

Trigonometric ratios are fundamental concepts in trigonometry and are essential for understanding right triangles. They are the ratios of the lengths of sides of a right triangle relative to one of its acute angles. The primary trigonometric ratios include:

• Sine (\(\sin\))
• Cosine (\(\cos\))
• Tangent (\(\tan\))

These ratios are specifically defined as follows:

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

These trigonometric functions are crucial for solving various problems involving right triangles and can also be extended to non-right triangles using the laws of sine and cosine.

Sine Function

The sine function is one of the core trigonometric functions used to relate an angle to the ratio of two sides of a right triangle. In a right triangle, the sine of an angle \(\theta\) can be defined by:
\[\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\]
This formula allows you to find a missing side or angle if you have at least one angle and one side. For instance, to calculate \(\sin 60^{\circ}\), we look at the corresponding special triangle.
It is important to remember that the sine function only applies to right triangles and acute angles within those triangles, specifically those less than \(90^{\circ}\). When dealing with the unit circle, the range of the sine function extends into greater angles, reaching values between \(-1\) and \(1\) for all real numbers.

30-60-90 Triangle

A \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle is a special type of right triangle that has specific side length ratios. It is called 'special' because of its consistent and predictable side lengths.

• The side opposite the \(30^{\circ}\) angle is the shortest and is often represented as \(x\).
• The side opposite the \(60^{\circ}\) angle is \(x\sqrt{3}\).
• The hypotenuse, which is the side opposite the right angle \(90^{\circ}\), is \(2x\).

Using these ratios simplifies the process of finding trigonometric functions. For example, to find \(\sin 60^{\circ}\), we use the ratio: opposite side over hypotenuse, resulting in \(\frac{x\sqrt{3}}{2x} = \frac{\sqrt{3}}{2}\). This pattern is fixed for all \(30^{\circ}-60^{\circ}-90^{\circ}\) triangles, making it easier for students to learn and apply trigonometric concepts in problems.