Pythagoras and basic trigonometry explained
Geometry and measures • Mensuration and calculation
Flashcards
Test your knowledge with interactive flashcards
Key concepts
What you'll likely be quizzed about
Pythagoras' theorem (definition, limits)
Pythagoras' theorem states that, in any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a^2 + b^2 = c^2. The theorem applies only when one angle is exactly 90° and when a, b denote the two legs and c denotes the hypotenuse. The relation gives an exact algebraic link between side lengths and enables direct computation of any one side when the other two are known.
Trigonometric ratios (definitions and orientation)
The primary trigonometric ratios for an acute angle θ in a right-angled triangle are sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent. The triangle orientation requires identification of the angle of interest and correct labelling of opposite, adjacent and hypotenuse before substituting into these ratios. The ratios provide a link from angle to side ratios and from side ratios to angle via inverse functions.
Exact values for sin and cos at standard angles
Exact values for sine and cosine at 0°, 30°, 45°, 60° and 90° follow from standard unit-circle and special-triangle constructions. The values for sin θ are 0, 1/2, √2/2, √3/2 and 1 at 0°, 30°, 45°, 60° and 90° respectively. The values for cos θ are 1, √3/2, √2/2, 1/2 and 0 at 0°, 30°, 45°, 60° and 90° respectively. These values enable exact evaluation of many trig expressions without decimal approximations.
Exact values for tangent at standard angles and limiting cases
Exact values for tangent follow from tan θ = sin θ / cos θ for angles where cos θ ≠ 0. The values at 0°, 30°, 45° and 60° are 0, 1/√3 (often written √3/3), 1 and √3 respectively. The tangent function is undefined at 90° because cos 90° = 0. Recognition of undefined cases prevents invalid algebraic steps and supports correct handling of vertical slopes in geometric problems.
Using Pythagoras and trig to find lengths
Pythagoras solves a missing side when two side lengths of a right-angled triangle are known. Trigonometric ratios solve missing sides when one side and one acute angle are known: multiply the known side by the appropriate ratio to obtain the unknown side. Combined use of Pythagoras and trig often simplifies problems where one side finds a second via trig and the third via Pythagoras.
Extending methods to general triangles and 3D figures
Right-angled triangle methods apply to many two- and three-dimensional problems by decomposition into right-angled components. Two-dimensional non-right triangles may use perpendicular construction or the sine and cosine rules when no right angle exists. Three-dimensional problems use right-angled cross-sections or projections to create right-angled triangles that yield lengths or angles in space. Careful labelling of dimensions and consistent identification of the relevant right-angled triangle ensure correct application.
Key notes
Important points to keep in mind