Right-angled triangles

A right-angled triangle has one angle of $90^\circ$. The side opposite that angle is the hypotenuse, and it is always the triangle’s longest side.

Once an acute angle has been chosen, the remaining sides are labelled opposite and adjacent. These labels change when a different angle is selected, so mark the angle first. A clear sketch should include the right angle, every known measurement and the quantity you need to find.

The three trigonometric ratios link an acute angle in a right-angled triangle with pairs of side lengths:

SOH–CAH–TOA is a useful memory aid, but the diagram—not the mnemonic—determines each side’s label.

Pick the ratio that contains both the known side and the required side. If the question involves the opposite side and the hypotenuse, for example, use sine. When it makes the working easier to follow, rearrange algebraically before substituting.

Use the matching inverse trigonometric function to find an angle:

$$\theta=\sin^{-1}\left(\frac{x}{z}\right),\qquad \theta=\cos^{-1}\left(\frac{y}{z}\right),\qquad \theta=\tan^{-1}\left(\frac{x}{y}\right)$$

Inverse functions undo the original trigonometric functions. The exponent $-1$ does not mean “take the reciprocal”. Use degree mode for calculations in this topic.

Non-right-angled triangles

In a non-right-angled triangle, each side must be labelled opposite its matching angle. This pairing is essential when using the sine rule.

The sine rule relates side lengths to their opposite angles:

Only two fractions are needed: use the known opposite pair and the pair containing the unknown. The syllabus does not require the ambiguous sine-rule case, in which the given information can produce two different triangles.

The cosine rule connects all three sides of a triangle with one angle:

$$c^2=a^2+b^2-2ab\cos C$$

Use this form to find a side when two sides and their included angle are known. If all three sides are known and an angle is required, rearrange it to

$$\cos C=\frac{a^2+b^2-c^2}{2ab}$$

In the cosine rule, the angle must sit opposite the isolated side. To find a different side, rotate all the lettering consistently; don’t change just one letter.

Two sides and their included angle can also be used to find the area of a triangle:

“Included” is crucial here: $C$ must be the angle between sides $a$ and $b$.

These rules support triangulation, a measurement method used to determine an inaccessible position or distance from a measured baseline and angles. It has long played an important role in surveying and map-making. The geometry also connects naturally with vectors in physics, where magnitudes and directions can form triangles.

Trigonometric ideas have a broad history. Results involving right triangles appear in early Chinese and Indian mathematical writing, while systematic trigonometric work developed prominently in Indian mathematics. This makes it difficult to attach a theorem to one person alone. A fair judgment about mathematical credit should consider surviving evidence, independent discovery, proof, transmission and the distinction between discovering a result and popularizing it.

Selecting and combining methods

Applied problems don’t usually tell you which rule to choose. Start by turning the written information into a labelled diagram. Add units, and mark any right angles, parallel horizontal lines or north directions. Never assume the sketch is drawn to scale.

For a right-angled triangle, Pythagoras' theorem gives

$$z^2=x^2+y^2$$

Use it when the problem involves two sides and no angle is needed. If an acute angle is involved, choose a right-triangle ratio. In a non-right-angled triangle, the sine rule works when you know an opposite side–angle pair. Use the cosine rule for three sides or two sides with their included angle, and use $K=\frac12ab\sin C$ to find area from two sides and the included angle.

More complicated diagrams can often be divided into two triangles. Keep the full calculator values for intermediate lengths and angles. Rounding too soon can noticeably change the final result.

Elevation and depression

An angle of elevation is measured upward from an observer's horizontal line of sight. An angle of depression is measured downward from the same horizontal. The horizontal through the observer is parallel to a horizontal through the target, so alternate angles often let you transfer the given angle into a right-angled triangle.

You may need to add or subtract the observer’s height from the vertical side found using trigonometry. Check the wording: the height calculated from the triangle isn’t automatically the object’s total height.

Bearings

A bearing is a clockwise angle measured from north that specifies a direction. Bearings are normally written using three digits, for example $065^\circ$. Draw a new north line at every location. Since north lines are parallel, corresponding and alternate-angle relationships can be useful.

A journey given through distances and bearings will usually form a triangle. Before applying trigonometry, convert the directional information into interior angles. For the final bearing, measure clockwise from north again rather than simply giving an interior angle.

Wider applications and mathematical assumptions

Triangulation is used in map-making, navigation and locating radio transmitters. Parallax is an apparent change in an object's direction caused by viewing it from two separated positions; the separation provides the baseline for a triangulation calculation. Scientific field studies use similar methods when a distance can’t be measured directly.

A scalar has magnitude but no direction, whereas a vector has both magnitude and direction. Distances are scalars; displacements and forces are vectors. As a result, vector diagrams in mechanics use the same triangle methods for sides and angles.

Large-scale surveys have also used triangulation to test models of Earth's curvature, including historical work linked to debates over gravitational theory between English and French scientists. Geometric calculations can settle questions about the physical world only if both the measurements and the chosen model are reliable.

Pythagoras' theorem shouldn’t be described as the isolated invention of one person. Versions of the relationship and its proofs developed in several mathematical cultures. Many valid proofs exist, including rearrangement, similar-triangle and area arguments. They show that different chains of reasoning can lead to mathematical certainty.

In plane, or Euclidean, geometry, a triangle satisfies $A+B+C=180^\circ$. On a positively curved surface such as a sphere, the sum can exceed $180^\circ$; in negatively curved geometry it can be less. Mathematical conclusions are certain relative to the stated definitions and assumptions, rather than independent of the geometry in which they are made.

Parts of a circle

An arc is a connected section of a circle's circumference. A sector is the region bounded by two radii and the arc joining them. The angle formed by the two radii is called the central angle.

At this level, central angles are measured in degrees; radians are not required. To find an arc length or sector area, take the fraction $\frac{\theta}{360^\circ}$ of the complete circle.

Arc length

The length of an arc is

Since $2\pi r$ gives the full circumference, the angle fraction picks out the required portion.

Sector area

The area of a sector is

In this formula, $\pi r^2$ gives the area of the complete circle. For a major sector, either subtract the minor central angle from $360^\circ$ or subtract the minor sector's result from the whole circle.

Physical experience supports these proportional formulas. Folding or dividing a circular object suggests that equal central angles cut off equal arcs and equal areas. Such experience may point towards the relationship, but definitions and deductive reasoning secure the mathematical claim. In mathematics, personal observation often leads to a conjecture. In empirical subjects, repeated observation may remain part of the evidence supporting the final claim.