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 both the unit circle and right triangle trigonometry.

5 Must Know Facts For Your Next Test

1. In a 30-60-90 triangle, the side opposite the 30-degree angle is half the length of the hypotenuse.
2. The side opposite the 60-degree angle is equal to the length of the hypotenuse multiplied by $\sqrt{3}$.
3. The side opposite the 90-degree angle is equal to the length of the hypotenuse.
4. The 30-60-90 triangle is commonly used to find the trigonometric ratios of special angles in the unit circle.
5. The 30-60-90 triangle is also useful in right triangle trigonometry to solve for unknown side lengths and angle measures.

Review Questions

• Explain how the 30-60-90 triangle is used to find the trigonometric ratios of special angles in the unit circle.
  • The 30-60-90 triangle is a special right triangle with unique side length relationships that can be used to determine the trigonometric ratios of the 30-degree, 60-degree, and 90-degree angles in the unit circle. Specifically, the side opposite the 30-degree angle is half the length of the hypotenuse, the side opposite the 60-degree angle is equal to the hypotenuse multiplied by $\sqrt{3}$, and the side opposite the 90-degree angle is equal to the length of the hypotenuse. These relationships allow you to easily calculate the sine, cosine, and tangent of these special angles in the unit circle.
• Describe how the properties of the 30-60-90 triangle can be used to solve right triangle trigonometry problems.
  • The unique side length relationships of the 30-60-90 triangle can be leveraged to solve a variety of right triangle trigonometry problems. If you know the length of one side of a 30-60-90 triangle, you can use the special ratios to determine the lengths of the other sides. For example, if you know the length of the hypotenuse, you can find the lengths of the other two sides by dividing the hypotenuse by 2 for the 30-degree side and multiplying the hypotenuse by $\sqrt{3}$ for the 60-degree side. Conversely, if you know the lengths of two sides, you can use the ratios to find the third side and solve for unknown angle measures.
• Analyze how the properties of the 30-60-90 triangle are connected to the broader concepts of unit circle trigonometry and right triangle trigonometry.
  • The 30-60-90 triangle serves as a critical link between the unit circle and right triangle trigonometry. Its unique angle and side length relationships allow you to easily translate between the trigonometric ratios in the unit circle and the side lengths and angle measures of a right triangle. This connection is essential for understanding how the unit circle and right triangle trigonometry are related and for seamlessly moving between the two concepts when solving problems. The 30-60-90 triangle provides a fundamental framework for applying trigonometric principles in a variety of contexts, making it a cornerstone of both unit circle and right triangle trigonometry.

Related terms

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

Special Right Triangles: Right triangles with specific angle and side length relationships, such as the 30-60-90 triangle and the 45-45-90 triangle.

Trigonometric Ratios: The ratios of the sides of a right triangle, including sine, cosine, and tangent, that are used to solve for unknown side lengths and angle measures.