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For a right triangle with legs and and hypotenuse , the Pythagorean theorem states:
The hypotenuse is opposite the right angle. We will use this theorem to derive exact side ratios for special right triangles.
Trigonometric functions begin as ratios of side lengths. A ratio compares two quantities, while a proportion states that two ratios are equal.
Suppose a vertical ruler and a vertical tree stand on the same level ground and their shadows are measured at the same time. Parallel sunlight makes the two height-shadow triangles similar. If the ruler is tall, its shadow is , and the tree's shadow is , let the tree height be :
The proportion works because the two triangles share the same two acute angles. Two shadows alone do not guarantee equal ratios.
Similar triangles have equal corresponding angles and proportional corresponding sides. Any one of these standard tests is sufficient to prove similarity:
| Test | Sufficient information |
|---|---|
| AA | Two pairs of corresponding angles are equal. The third pair is then equal automatically. |
| SSS | All three pairs of corresponding sides are proportional. |
| SAS | Two pairs of corresponding sides are proportional and their included angles are equal. |
Similar triangles have the same shape even when their sizes differ. Every corresponding length is multiplied by the same scale factor.
Use the common scale factor to find unknown lengths or preserve proportions:
Height measurement: Shadows or sight lines can form similar triangles for objects that are difficult to measure directly.
Scale drawings: A plan or model preserves proportions while every length is enlarged or reduced by one scale factor.
Geometry and construction: Indirect measurements can be checked through corresponding angles and proportional sides.
Trigonometric ratios compare the sides of a right triangle relative to a chosen acute angle.
Imagine a right triangle with an angle (theta). We name its sides as:
From these three sides, we obtain six basic trigonometric ratios. For an acute angle in a nondegenerate right triangle, all three side lengths are positive, so every ratio below is defined:
Sine ():
Cosine ():
Tangent ():
Cosecant ():
Secant ():
Cotangent ():
Two right triangles with the same acute angle, for example , also share the right angle. The AA test therefore proves that they are similar. Scaling every side by the same factor leaves each quotient of corresponding side lengths unchanged.
Simple example:
If we calculate for both triangles:
Although the triangles have different sizes, has the same value. The same reasoning applies to the other side ratios.
On the unit circle, the point corresponding to has coordinates . This definition extends sine and cosine beyond the acute angles of a right triangle.
The right-triangle ratios and the unit-circle coordinates agree for acute angles, so the same values can be used in problems with triangles of different sizes.
Trigonometry connects angles with lengths when measuring shadows and roof slopes. On the unit circle, sine and cosine also repeat after one turn. That repetition is used to model waves, while celestial navigation requires geometry on Earth's surface.
For nearby vertical objects on level ground, height divided by shadow length is the same when the shadows are measured at the same time. This follows from parallel sunlight and similar triangles.
More than ago, Eratosthenes estimated Earth's circumference by comparing the Sun's angle at two locations and combining the angular difference with the distance between those locations. Without the distance, the angular difference alone cannot determine the circumference.
In celestial navigation, an observer measures a celestial body's altitude above the horizon and records the observation time. The U.S. Naval Observatory provides astronomical data for turning such observations into lines of position. One observation constrains the possible vessel positions. Several observations, together with time and vessel-motion data, are needed to determine a position.
Suppose a roof makes an angle of with the horizontal, and the horizontal run from its edge to its peak is . The rafter is the hypotenuse, so its length follows from cosine:
The rafter is longer than the horizontal run, as the hypotenuse must be the longest side. This calculation excludes any overhang beyond the roof edge.
Sound waves, radio waves, and alternating electric currents can be modeled using sine and cosine functions. A wave repeats, so an angle that grows past a full turn describes the next cycle rather than a new direction. Sine and cosine carry that repetition for every periodic signal.
Choose nearby vertical objects on level ground and measure their shadows at nearly the same time. This keeps the solar angle effectively the same.
Measure a reference height and its horizontal shadow length .
Calculate the common ratio .
Measure another object's shadow length and estimate its height with .
If desired, calculate the Sun's elevation angle with .
For example, if a reference object casts a shadow, then . An object with an shadow has the estimated height
If the ground slopes, an object is not vertical, or the measurement times differ substantially, this simple model no longer applies without adjustment.
Published: . Updated: .
