Chapter 9: Pythagoras' Theorem and Trigonometry

Vocabulary practice flashcards covering key definitions, big ideas, trigonometric ratios, bearings, and angles of elevation and depression from Chapter 9.

Proportional reasoning

The mathematical understanding that ratios of side lengths in right-angled triangles remain constant ($\sin(\theta)$, $\cos(\theta)$, and $\tan(\theta)$), allowing triangles to be scaled up or down while maintaining the same trigonometric relationships.

Spatial reasoning

The ability to visualise shapes, mentally rotate them, and see connections between different parts of a figure to understand relationships between sides and angles.

Algebraic reasoning

Using letters and symbols to represent quantities to allow for generalisations, such as variables in Pythagoras' theorem ($a^2 + b^2 = c^2$) representing any real number side lengths that fit a right-angled triangle.

Geometric properties

Foundational concepts such as congruence and similarity; for example, similar triangles having the same shape but not necessarily the same size, which underpins why trigonometric ratios remain constant.

Number sense

An intuitive understanding of numbers, their magnitudes, and their relationships, such as recognizing that the hypotenuse is always the longest side in a right-angled triangle.

Exact form

A number that is not expressed as a decimal approximation, but is instead left in fraction or surd form (e.g., $\sqrt{61}$).

Pythagorean triples

Collections or triples of whole numbers that always produce the side lengths of a right-angled triangle.

Hypotenuse

The longest side of a right-angled triangle, which is always located opposite the right angle.

Pythagoras' theorem

A geometric relation establishing that in any right-angled triangle, the square of the hypotenuse ($c$) equals the sum of the squares of the two shorter sides ($a$ and $b$), expressed as $c^2 = a^2 + b^2$.

Adjacent side

The side of a right-angled triangle that forms an arm of the reference angle other than the hypotenuse, positioned directly next to the reference angle.

Opposite side

The side of a right-angled triangle that is situated directly across from or facing the reference angle.

Sine ratio

A trigonometric ratio defined as the length of the opposite side divided by the length of the hypotenuse: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{O}{H}$.

Cosine ratio

A trigonometric ratio defined as the length of the adjacent side divided by the length of the hypotenuse: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{A}{H}$.

Tangent ratio

A trigonometric ratio defined as the length of the opposite side divided by the length of the adjacent side: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{O}{A}$.

Subject of a formula

The variable that is isolated entirely by itself on one side of the equals sign in an algebraic equation.

Substitution

The algebraic process of replacing a variable or unknown pronumeral with a given numerical value.

Inverse operations

Mathematical operations that reverse each other; addition and subtraction are inverses, as are multiplication and division.

Inverse trigonometric ratios

Mathematical operations ($\sin^{-1}$, $\cos^{-1}$, and $\tan^{-1}$) used to determine unknown angles in a right-angled triangle when two side lengths are known.

Acute angle

An angle that is greater than $0^\circ$ but less than $90^\circ$ in size.

Due bearing

An exact cardinal direction, such as directly north, east, west, or south.

True bearing

A direction measured as an angle clockwise from north, written using three digits followed by the letter $\text{T}$ (e.g., $072^\circ\text{ T}$).

Compass bearing

A direction expressed as an acute angle measured from north or south towards east or west, written with a starting direction ($\text{N}$ or $\text{S}$), an acute rotation angle, and a direction of rotation ($\text{E}$ or $\text{W}$).

Opposite directions (bearings)

Bearings that point directly away from each other and differ by $180^\circ$; calculated as $\theta + 180^\circ$ if $0^\circ \le \theta < 180^\circ$, and $\theta - 180^\circ$ if $180^\circ \le \theta < 360^\circ$.

Line of vision

The implied straight line along which an observer looks directly at an object, also referred to as the direct distance.

Horizontal line

A line that is parallel with the natural horizon.

Vertical line

A line that is perpendicular (at a $90^\circ$ angle) to the horizon.

Angle of elevation

The angle formed between the horizontal line and the observer's line of vision when looking upwards at an object.

Angle of depression

The angle formed between the horizontal line and the observer's line of vision when looking downwards at an object.

Clinometer

An instrument used in land surveying to measure angles between objects, including angles of elevation and depression.