Lines drawn to horizontal to form triangle ratios (reference triangles, angles). Axes, quadrants, abscissa, ordinate, and distance labeled.

Reference Angles/Triangles Formed by Angles in Quadrants With Labels

Lines drawn to horizontal to form triangle ratios (reference triangles, angles). Axes, quadrants, abscissa,…

Circle with 36 degree angles marked. This diagram can be used with the following trig problem: Locate the centers of the holes B and C by finding the distance each is to the right and above the center O. The radius of the circle is 1.5 inches. Compute correct to three decimal places.

Circle With 36 degree Angles and Radius 1.5 in.

Circle with 36 degree angles marked. This diagram can be used with the following trig problem: Locate…

Circle modeling the earth. O is the center of the earth, r the radius of the earth, and h the height of the point P above the surface; it is required to find the distance from the point P to the horizon at A.

Circle With Center o and Radius r with point P

Circle modeling the earth. O is the center of the earth, r the radius of the earth, and h the height…

Circle with chord AB=2 ft. and radius OA = 3 ft.. Triangle AOC is a right triangle. Angle AOC=half angle AOB, and the central angle AOB has the same measure as the arc AnB.

Circle With a Chord of 2 ft. and a Radius of 3 ft.

Circle with chord AB=2 ft. and radius OA = 3 ft.. Triangle AOC is a right triangle. Angle AOC=half angle…

Illustration of a man using a cross-staff, a navigation tool used largely at sea. It was originally created for use in land navigation and finds latitude by measuring the distance of a star above the horizon. It cannot find longitude. The man in the drawing has the cross-staff positioned with the rod on his cheek. There is a star above him. He is wearing a wide, floppy-brimmed hat.

A Cross-Staff Used for Navigation at Sea

Illustration of a man using a cross-staff, a navigation tool used largely at sea. It was originally…

Illustration of blueprint used by highway engineers to widen the pavement on the inside of the curve of a road.

Curve in Pavement of Road

Illustration of blueprint used by highway engineers to widen the pavement on the inside of the curve…

To determine the distance between two points A, B given by their projections.

Point Distance

To determine the distance between two points A, B given by their projections.

An illustration of military personnel measuring distance.

Military Distance Measurement

An illustration of military personnel measuring distance.

"The radiation of light, or the intensity of light is inversely proportional to the square of the distance from the light source." —Croft 1917

Light Intensity

"The radiation of light, or the intensity of light is inversely proportional to the square of the distance…

"Another illustration of how luminous density varies inversely as the square of the distance." —Croft 1917

Light Intensity Versus Distance

"Another illustration of how luminous density varies inversely as the square of the distance." —Croft…

"A card placed at point A receives normal light. At point B the card receives four times as much light, but only at 1/4 intensity. At point C nine times the area covered, but only 1/9 the intensity." —Quackenbos 1859

Light Intensity at Different Distances

"A card placed at point A receives normal light. At point B the card receives four times as much light,…

"Let a be the distance at which an object can be see distinctly, and b, the distance at which the same object is seen through the lens, and supose tthe distance of a from the eye, be twice that of b. Then because the object is at half the distance that it was before, it will appear twice as large...." -Comstock 1850

Single Microscope

"Let a be the distance at which an object can be see distinctly, and b, the distance at which the same…

"Relative distance of the Planets. Having now given a short account of each planet composing the solar system, the relative situation of their several orbits, with the exception of those of the Asteroids, are shown in this figure. The orbits are marked by the signs of each planet, of which the first, or that nearest the Sun, is Mercury, the next Venus, the third the Earth, the fourth Mars then come those of the Asteroids, then Jupiter, then Saturn and lastly Herschel." —Comstock, 1850

Planet Distance

"Relative distance of the Planets. Having now given a short account of each planet composing the solar…

Illustration modeling the path of a projectile.

Path of Projectile

Illustration modeling the path of a projectile.

Roof with a slope of 30 degrees. Angle theta is the inclination to the horizontal of the line AB, drawn in the roof and making an angle of 35 degrees with the line of greatest slope.

Roof With 30 degree Inclination for Trigonometry Triangle Problems

Roof with a slope of 30 degrees. Angle theta is the inclination to the horizontal of the line AB, drawn…

These are skew bevel wheels, made of a hyperboloid of revolution around an axis.

Sliding Contact

These are skew bevel wheels, made of a hyperboloid of revolution around an axis.

An illustration of a square and triangles within the square. This is an example of a problem that can be used to fine the distances of two inaccessible objects without measuring elevation and whether on a horizontal plane or not.

Square

An illustration of a square and triangles within the square. This is an example of a problem that can…

Two set squares, whose sides are 3,4, and 5 in., are placed so that their 4-in. sides and right angles coincide, and the angle between the 3-in. sides is 50 degrees. Theta, Ǝ is the angle between the longest sides.

Two Squares With Sides of Lengths 3,4,5 Placed at Right Angles to Each Other

Two set squares, whose sides are 3,4, and 5 in., are placed so that their 4-in. sides and right angles…

A team of surveyors in the country.

Surveying

A team of surveyors in the country.

Illustration of oblique triangle used to find distance across a river.

Oblique Triangle for Distance Across a River

Illustration of oblique triangle used to find distance across a river.

Illustration of oblique triangle used to find distance across a river.

Oblique Triangle for Distance Across a River

Illustration of oblique triangle used to find distance across a river.

Illustration of oblique triangle used to find distance across a lake.

Oblique Triangle for Distance Across Lake

Illustration of oblique triangle used to find distance across a lake.

Inclined plane forming right triangle showing the velocity of a body sliding a distance,s, down a smooth horizontal plane.

Inclined Plane Forming Right Triangle

Inclined plane forming right triangle showing the velocity of a body sliding a distance,s, down a smooth…

Illustration of similar triangles used to find distance across a lake.

Similar Triangles for Distance Across Lake

Illustration of similar triangles used to find distance across a lake.

Illustration of similar triangles used to find distance across a stream

Similar Triangles for Distance Across Stream

Illustration of similar triangles used to find distance across a stream

"The wedge is a movable inclined plane, and is used for moving a great weight a short distance. A common method is shown here." —Hallock 1905

Wedge

"The wedge is a movable inclined plane, and is used for moving a great weight a short distance. A common…