Knowledge and use of 30-60-90 and 45-45-90 Triangles

Key Features of Special Triangles

The 30-60-90 triangle and the 45-45-90 triangle are two special kinds of right-angled triangles.
Understanding these triangles allow simplified calculations in certain geometrical problems or trigonometric applications.

30-60-90 Triangle

The sides of a 30-60-90 triangle are in ratio of √1 : √3 : 2.
The length of the hypotenuse is twice the length of the shorter leg (which is opposite the 30-degree angle).
The length of the longer leg (which is opposite the 60-degree angle) is √3 times the length of the shorter leg.

45-45-90 Triangle

In a 45-45-90 triangle, the sides are in the ratio of 1 : 1 : √2.
Both the legs are equal in length as the angles opposite them (which are 45 degrees each) are equal.
The length of the hypotenuse is √2 times the length of each leg.

Trigonometric Ratios in Special Triangles

Understanding these triangles also allows simplified computation of certain trigonometric values.

30-60-90 Triangle

For the 30-degree angle: sin 30° = 1/2, cos 30° = √3/2, tan 30° = √3/3.
For the 60-degree angle: sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3.

45-45-90 Triangle

Since the two non-right angles are both 45 degrees: sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1.

By recognising these patterns in 30-60-90 and 45-45-90 triangles, you can save significant effort and time when solving geometry problems or performing calculations that involve trigonometry.