Special Right Triangles: 45-45-90 and 30-60-90 Triangles

Special Right Triangles

45-45-90 Triangles

Triangles with angles of 45°, 45°, and 90° are called isosceles right triangles. These triangles have unique properties that allow for quick calculation of side lengths and trigonometric ratios without the Pythagorean Theorem.

• Legs: The two legs are always the same length.

• Hypotenuse: The hypotenuse is always √2 times the length of a leg.

Key Formula:

If each leg has length $x$, then the hypotenuse is $x\sqrt{2}$.

$\text{Hypotenuse} = x\sqrt{2}$

Example:

• If each leg is 11, then hypotenuse $= 11\sqrt{2}$.

• If hypotenuse is 12, then each leg $= \dfrac{12}{\sqrt{2}} = 6\sqrt{2}$ (rationalize denominator if needed).

Practice Problem

• Given a 45-45-90 triangle with leg $x$, hypotenuse is $x\sqrt{2}$.

Common Trigonometric Functions for 45-45-90 Triangles

The trigonometric ratios for 45-45-90 triangles are consistent and can be memorized for quick reference.

Function	Ratio	Value
sin 45°	Opp/Hyp	$\dfrac{x}{x\sqrt{2}} = \dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{2}}{2}$
cos 45°	Adj/Hyp	$\dfrac{x}{x\sqrt{2}} = \dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{2}}{2}$
tan 45°	Opp/Adj	$\dfrac{x}{x} = 1$
csc 45°	Hyp/Opp	$\dfrac{x\sqrt{2}}{x} = \sqrt{2}$
sec 45°	Hyp/Adj	$\dfrac{x\sqrt{2}}{x} = \sqrt{2}$
cot 45°	Adj/Opp	$\dfrac{x}{x} = 1$

30-60-90 Triangles

Triangles with angles of 30°, 60°, and 90° are called 30-60-90 triangles. These triangles have side lengths in a fixed ratio, making calculations straightforward.

• Shortest leg (opposite 30°): $x$

• Longer leg (opposite 60°): $x\sqrt{3}$

• Hypotenuse (opposite 90°): $2x$

Key Formulas:

• Longer leg $= x\sqrt{3}$

• Hypotenuse $= 2x$

Example:

• If the shortest leg is 4, then longer leg $= 4\sqrt{3}$, hypotenuse $= 8$.

• If hypotenuse is 13, then shortest leg $= 6.5$, longer leg $= 6.5\sqrt{3}$.

Practice Problem

• Given a 30-60-90 triangle with shortest leg $x$, longer leg is $x\sqrt{3}$, hypotenuse is $2x$.

Common Trigonometric Functions for 30-60-90 Triangles

The trigonometric ratios for 30-60-90 triangles are also fixed and useful for solving problems quickly.

Function	Ratio	Value (30°)	Value (60°)
sin	Opp/Hyp	$\dfrac{x}{2x} = \dfrac{1}{2}$	$\dfrac{x\sqrt{3}}{2x} = \dfrac{\sqrt{3}}{2}$
cos	Adj/Hyp	$\dfrac{x\sqrt{3}}{2x} = \dfrac{\sqrt{3}}{2}$	$\dfrac{x}{2x} = \dfrac{1}{2}$
tan	Opp/Adj	$\dfrac{x}{x\sqrt{3}} = \dfrac{1}{\sqrt{3}} = \dfrac{\sqrt{3}}{3}$	$\dfrac{x\sqrt{3}}{x} = \sqrt{3}$
csc	Hyp/Opp	$\dfrac{2x}{x} = 2$	$\dfrac{2x}{x\sqrt{3}} = \dfrac{2}{\sqrt{3}} = \dfrac{2\sqrt{3}}{3}$
sec	Hyp/Adj	$\dfrac{2x}{x\sqrt{3}} = \dfrac{2}{\sqrt{3}} = \dfrac{2\sqrt{3}}{3}$	$\dfrac{2x}{x} = 2$
cot	Adj/Opp	$\dfrac{x\sqrt{3}}{x} = \sqrt{3}$	$\dfrac{x}{x\sqrt{3}} = \dfrac{1}{\sqrt{3}} = \dfrac{\sqrt{3}}{3}$

Summary Table: Side Ratios for Special Right Triangles

Triangle	Angle Measures	Side Ratios
45-45-90	45°, 45°, 90°	$1 : 1 : \sqrt{2}$
30-60-90	30°, 60°, 90°	$1 : \sqrt{3} : 2$

Additional Info:

• These triangles are frequently used in trigonometry, geometry, and calculus for simplifying calculations and solving problems involving right triangles.

• Memorizing the side ratios and trigonometric values for these triangles is highly recommended for Precalculus students.