Year 9 Trigonometry Theory Summary

Comprehensive vocabulary flashcards covering Year 9 trigonometry concepts, including triangle labelling, Pythagoras' theorem, trigonometric ratios (SOH CAH TOA), inverse functions, and angles of elevation and depression.

Hypotenuse

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

Opposite side

The side in a right-angled triangle that is positioned directly across from the chosen reference angle $\theta$.

Adjacent side

The side in a right-angled triangle that is positioned next to the chosen reference angle $\theta$ and is not the hypotenuse.

Right-angled triangle side labelling

The convention of identifying sides relative to a reference angle $\theta$, where the hypotenuse ($H$) is opposite the right angle, the opposite side ($O$) is across from $\theta$, and the adjacent side ($A$) is next to $\theta$.

Pythagoras' theorem

The geometric formula $a^2 + b^2 = c^2$, where $c$ is the hypotenuse, used to find an unknown side length when two side lengths are known.

Right angle test

A procedure to verify if a triangle contains a right angle by checking whether the square of the longest side equals the sum of the squares of the two shorter sides.

SOH CAH TOA

A mnemonic representing the primary trigonometric ratios: Sine = Opposite $\div$ Hypotenuse, Cosine = Adjacent $\div$ Hypotenuse, and Tangent = Opposite $\div$ Adjacent.

Sine ratio ($\sin(\theta)$)

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

Cosine ratio ($\cos(\theta)$)

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

Tangent ratio ($\tan(\theta)$)

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

Unknown in the numerator

A trigonometric equation setup where the unknown side is in the top position of the fraction, solved by multiplying the known denominator by the trigonometric ratio (e.g., $O = H \times \sin(\theta)$).

Unknown in the denominator

A trigonometric equation setup where the unknown side is underneath the fraction, solved by rearranging before using a calculator and dividing the numerator by the trigonometric ratio (e.g., $H = O \div \sin(\theta)$).

Inverse trigonometric function

A mathematical function ($\sin^{-1}$, $\cos^{-1}$, or $\tan^{-1}$) used to determine an unknown angle when two sides are known, where the superscript $-1$ denotes the inverse operation rather than a reciprocal.

Degrees and minutes

A system of angular subdivision where $1^\circ = 60'$; if rounding results in $60'$, $1^\circ$ is carried over and $0'$ is recorded.

Angle of elevation

An angle measured upwards from a horizontal line to an object or point.

Angle of depression

An angle measured downwards from a horizontal line to an object or point.

Elevation and depression equality

The geometric principle that on level ground, the matching angle of elevation equals the angle of depression because the horizontal reference lines are parallel.

Acute angles in a right-angled triangle

The two non-right angles in any right-angled triangle, which always sum to $90^\circ$.