Special Right Triangles and Unit Circle

Overview

This lecture covers the properties of special right triangles (45-45-90 and 30-60-90) and their connection to the unit circle, including how to find points using right triangle relationships.

Special Right Triangles

• Special right triangles only apply to triangles with a 90° angle.
• The Pythagorean theorem: (a^2 + b^2 = c^2) applies to these triangles.

45-45-90 Triangles

• An isosceles right triangle has two 45° angles and two equal legs.
• If each leg is (x), the hypotenuse is (x\sqrt{2}).
• To find a leg from the hypotenuse, divide by (\sqrt{2}).
• Both legs always have equal length.

30-60-90 Triangles

• This triangle has angles 30°, 60°, and 90°; the shortest leg is opposite 30°.
• The hypotenuse is twice the length of the shortest leg ((2x)).
• The longer leg is (x\sqrt{3}), where (x) is the short leg.
• To find the long leg, multiply the short leg by (\sqrt{3}).
• These ratios can be derived using equilateral triangle properties.

Example Problems

• For a 45-45-90 triangle with hypotenuse 26, each leg is (13\sqrt{2}).
• For a 30-60-90 triangle with short leg 6, hypotenuse is 12, and long leg is (6\sqrt{3}).
• The Pythagorean theorem checks your calculations.

The Unit Circle

• The unit circle has radius 1; every radius is length 1.
• Key angles are 45°, 30°, and 60°, corresponding to special triangles.
• For 45°, the coordinates are ((\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})) from the 45-45-90 triangle.
• For 30° or 60°, use the 30-60-90 triangle: coordinates like ((\frac{1}{2}, \frac{\sqrt{3}}{2})) or its negatives.
• Negative coordinates occur when points are left/down from the origin.

Key Terms & Definitions

• Hypotenuse — The side opposite the right angle in a triangle, the longest side.
• Leg — Either of the two shorter sides of a right triangle.
• Unit Circle — A circle with radius 1 centered at the origin of a coordinate plane.
• Pythagorean Theorem — For right triangles, (a^2 + b^2 = c^2).
• Radians — A unit of measuring angles; (\pi) radians = 180°.

Action Items / Next Steps

• Practice finding missing sides in 45-45-90 and 30-60-90 triangles.
• Fill in the remaining coordinates and angles (degrees and radians) on the unit circle.
• Draw special right triangles for each point on the unit circle as practice.
• Bring completed exercises to class for review.