30-60-90 Triangle

Definition

A 30-60-90 triangle is a right triangle with angles of 30 degrees, 60 degrees, and 90 degrees. This special type of right triangle has unique properties that make it useful in various trigonometric applications.

5 Must Know Facts For Your Next Test

1. The sides of a 30-60-90 triangle have a unique ratio of 1:'$\sqrt{3}$:2', where the shortest side is 1 unit, the medium side is '$\sqrt{3}$' units, and the longest side is 2 units.
2. The sine of the 30-degree angle is '$\frac{1}{2}$', the cosine is '$\frac{\sqrt{3}}{2}$', and the tangent is '$\frac{1}{\sqrt{3}}$'.
3. Trigonometric identities, such as the Pythagorean identity, can be easily verified using the side lengths of a 30-60-90 triangle.
4. 30-60-90 triangles are commonly used in engineering, architecture, and other fields to simplify calculations and problem-solving.
5. The properties of a 30-60-90 triangle can be used to derive formulas for the area and perimeter of the triangle.

Review Questions

• Explain how the side lengths of a 30-60-90 triangle are related and how this can be used to simplify trigonometric calculations.
  • The side lengths of a 30-60-90 triangle have a unique ratio of 1:'$\sqrt{3}$:2', where the shortest side is 1 unit, the medium side is '$\sqrt{3}$' units, and the longest side is 2 units. This ratio can be used to easily calculate the trigonometric ratios for the 30-degree and 60-degree angles, as the sine of the 30-degree angle is '$\frac{1}{2}$', the cosine is '$\frac{\sqrt{3}}{2}$', and the tangent is '$\frac{1}{\sqrt{3}}$. This makes 30-60-90 triangles particularly useful in simplifying trigonometric calculations and verifying trigonometric identities.
• Describe how the properties of a 30-60-90 triangle can be used to derive formulas for the area and perimeter of the triangle.
  • Given the unique side length ratio of a 30-60-90 triangle, it is possible to derive formulas for the area and perimeter of the triangle. The area of a 30-60-90 triangle can be calculated as '$\frac{1}{2} \times 1 \times \sqrt{3} = \frac{\sqrt{3}}{2}$', where the base is 1 unit and the height is '$\sqrt{3}$' units. The perimeter can be calculated as 1 + '$\sqrt{3}$' + 2 = '3 + \sqrt{3}$', using the side lengths of 1, '$\sqrt{3}$', and 2 units. These formulas can be used to quickly determine the area and perimeter of any 30-60-90 triangle without having to perform extensive calculations.
• Evaluate how the properties of a 30-60-90 triangle can be used to simplify the verification of trigonometric identities.
  • The unique side length ratios of a 30-60-90 triangle allow for the easy verification of trigonometric identities. For example, the Pythagorean identity, '$\sin^2 \theta + \cos^2 \theta = 1$', can be readily verified using the trigonometric ratios of a 30-60-90 triangle. Substituting the values of '$\sin 30^\circ = \frac{1}{2}$' and '$\cos 30^\circ = \frac{\sqrt{3}}{2}$' into the identity, we get '$\left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 = 1$', which is true. This simplification process can be applied to various other trigonometric identities, making the 30-60-90 triangle a powerful tool for verifying and understanding these fundamental relationships.

Related terms

Right Triangle: A triangle with one 90-degree angle, also known as a right angle.

Trigonometric Ratios: The ratios of the sides of a right triangle, such as sine, cosine, and tangent.

Trigonometric Identities: Equations that relate the trigonometric ratios to one another, such as '$\sin^2 \theta + \cos^2 \theta = 1$.'