Trigonometric Ratios Module – Key Vocabulary

Vocabulary flashcards summarizing the main theorems, ratios, and properties covered in the Trigonometric Ratios Module review.

Geometric Mean

A value x such that x = √(ab); in right-triangle altitudes, the altitude to the hypotenuse is the geometric mean of the two hypotenuse segments.

Altitudes in Right Triangles

An altitude drawn to the hypotenuse creates two smaller right triangles that are similar to the original triangle and to each other.

Special Right Triangle: 30-60-90

A right triangle whose acute angles measure 30° and 60°; its side lengths are in the fixed ratio 1 : √3 : 2.

Special Right Triangle: 45-45-90

An isosceles right triangle; legs are congruent and the hypotenuse is √2 times a leg.

30-60-90 Side Ratio

Shortest leg = 1, longer leg = √3, hypotenuse = 2 (all proportional).

45-45-90 Side Ratio

Leg : Leg : Hypotenuse = 1 : 1 : √2.

Hypotenuse Rule for 30-60-90

The hypotenuse is twice the length of the shorter (30°) leg.

Longer Leg Rule for 30-60-90

The side opposite 60° equals the shorter leg multiplied by √3.

Tangent Ratio

For acute angle θ in a right triangle, tan θ = opposite ⁄ adjacent.

Sine Ratio

For acute angle θ, sin θ = opposite ⁄ hypotenuse.

Cosine Ratio

For acute angle θ, cos θ = adjacent ⁄ hypotenuse.

Law of Sines

In any triangle, a ⁄ sin A = b ⁄ sin B = c ⁄ sin C; relates sides and opposite angles.

Law of Cosines

For any triangle, c² = a² + b² – 2ab cos C (and analogous forms for other sides).

SSA Ambiguous Case

With two sides and a non-included acute angle, zero, one, or two distinct triangles may exist.

Distance Formula (2D)

Distance between (x₁,y₁) and (x₂,y₂) is √[(x₂–x₁)² + (y₂–y₁)²].

Distance Formula (3D)

Distance between (x₁,y₁,z₁) and (x₂,y₂,z₂) is √[(x₂–x₁)² + (y₂–y₁)² + (z₂–z₁)²].

Rectangle Diagonal Properties

Diagonals are congruent and bisect each other; each midpoint divides a diagonal into two equal segments.

Triangle Area Formula

Area = ½ (base)(height); usable with legs of right triangles or an altitude drawn to the base.

Using Tangent to Find Height

Height = (adjacent side)·tan θ when θ is the angle of elevation from the end of the adjacent side.

Opposite, Adjacent, Hypotenuse

Relative side names in a right triangle with respect to a chosen acute angle.

Angle of Elevation

The angle formed by a horizontal line and the line of sight looking upward to an object.

Geometric Mean in Right-Triangle Segments

In a right triangle, the altitude to the hypotenuse is the geometric mean of the two hypotenuse segments, and each leg is the geometric mean of the hypotenuse and its adjacent segment.

Surveying with Law of Sines

By measuring a baseline and two angles, the opposite side (e.g., width of a lake) can be found using the Law of Sines.

Finding Height with Sine and Ladder

Given ladder length L and angle θ with the ground, ladder height on the wall is L·sin θ.

Altitude BD in 30-60-90 Example

For a right triangle with hypotenuse 20 and 30-60 angles, altitude to the hypotenuse equals 10√3, using special-triangle ratios.