Pythagoras' Theorem and Trigonometry

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infoWhy this? Pythagoras' Theorem and trigonometry provide complementary methods for solving right-angled triangles. They connect sides, angles, coordinates, and real measurements, developing modelling skills and precise spatial reasoning.

scheduleWhy now? Year 8 introduced Pythagoras' Theorem, and subsequent algebra and coordinate work strengthened rearrangement and graph skills. We can now generalise distance calculations and introduce sine, cosine, tangent, and their inverses before applying them in more complex geometry.

Pythagoras’ Theorem states that in a right-angled triangle with legs `a` and `b` and hypotenuse `c`, `a^2 + b^2 = c^2`.
Pythagoras’ Theorem can only be used in right-angled triangles; using it in a non-right-angled triangle gives incorrect results.
The hypotenuse is the longest side of a right-angled triangle and is always opposite the right angle.
If `a^2 + b^2 = c^2`, then `c = \sqrt{a^2 + b^2}`; if `a^2 + b^2 = c^2`, then a leg length can be found using `a = \sqrt{c^2 - b^2}` (or similar).
On a coordinate grid, the horizontal change between points is `\Delta x = x_2 - x_1` and the vertical change is `\Delta y = y_2 - y_1`.
The distance between two points on a coordinate grid is `d = \sqrt{(\Delta x)^2 + (\Delta y)^2}` because the line segment forms the hypotenuse of a right-angled triangle with legs `|\Delta x|` and `|\Delta y|`.
A square root gives a non-negative answer for a length, so distances and side lengths are not negative.
In this unit, SOHCAHTOA is used to relate the sides and an acute angle of a right-angled triangle.
For an acute angle `\theta` in a right-angled triangle: `\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}`, `\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}`, and `\tan \theta = \frac{\text{opposite}}{\text{adjacent}}`.
‘Opposite’ and ‘adjacent’ are defined relative to the chosen angle `\theta`, but the hypotenuse stays the same (opposite the right angle).
To find an angle from a ratio, use inverse trigonometry: `\theta = \sin^{-1}(\frac{\text{opp}}{\text{hyp}})`, `\theta = \cos^{-1}(\frac{\text{adj}}{\text{hyp}})`, or `\theta = \tan^{-1}(\frac{\text{opp}}{\text{adj}})`.
When using a calculator for trigonometry in school geometry, angles are usually measured in degrees, so the calculator must be in degree mode.
In many real-life worded problems, a diagram can be modelled as a right-angled triangle by identifying a horizontal and a vertical (or perpendicular) length and the sloping length as the hypotenuse.
Lengths should be given with appropriate units (for example, cm, m) and rounding should match the question, often to a stated number of decimal places or significant figures.

rocket_launchYou must be able to

Identify and label the right angle, hypotenuse, opposite and adjacent correctly on a given right-angled triangle for a specified angle `\theta`.
Use Pythagoras’ Theorem to calculate a missing side length by substituting known values into `a^2 + b^2 = c^2` and rearranging, showing clear working and a final square root evaluation.
Find the length of a line segment between two points on a coordinate grid by calculating `\Delta x` and `\Delta y` and then applying `d = \sqrt{(\Delta x)^2 + (\Delta y)^2}`.
Translate a worded problem into a right-angled triangle diagram, choosing and labelling known and unknown sides/angles so a suitable method (Pythagoras, sine, cosine, or tangent) is clear.
Select and apply the correct trigonometric ratio (sin, cos, or tan) to find a missing side length, setting up the equation with correct numerator and denominator before rearranging.
Use inverse trigonometric functions to find a missing acute angle, substituting a correct ratio and giving the angle in degrees to an appropriate level of accuracy.
Check the reasonableness of answers by comparing to known constraints (for example, the hypotenuse must be the longest side; an acute angle must be between `0^\circ` and `90^\circ`).
Present final answers with correct units and appropriate rounding, and include a brief concluding statement matching what the question asked (for example, ‘The distance is … m’).

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