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Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. At 45° increments, the angles are given in both radian and degree measure. At each angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled In 45° Increments With Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. The circle is marked and labeled in both radians and degrees at all quadrantal angles and angles that have reference angles of 30°, 45°, and 60°. At each angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled With Special Angles And Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. The circle is marked and labeled in both radians and degrees at all quadrantal angles and angles that have reference angles of 30°, 45°, and 60°. At each angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled With Special Angles And Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. At 30° increments, the angles are given in both radian and degree measure. At each angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled In 30° Increments With Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Leaves - simple; alternate; edge sharply and often doubly toothed. Outline - oval or egg-shaped, or inversely egg-shaped; always one-sided. Apex - taper-pointed. Base - rounded, or slightly heart-shaped, rarely pointed. Leaf/Stem - about one quarter inch long. Buds - smooth. Leaf - usually two to five inches long, and one and a half to two and a half wide; somewhat downy when young, afterward roughish below; above, either rough in one direction, or (especially if taken from the ends of the long branches) smooth and shining. Ribs - prominent and straight. Bark - of the branches not marked with "corky ridges"; branchlets, smooth. Seeds - flat egg-shaped or oval, winged and fringed all around. Last of May. Found - northward to Southern Newfoundland; southward to Florida; westward to the Black Hills of Dakota. Toward the western and southwestern limits it is found only in the river-bottom lands. General Information - One of the very noblest of American trees, eighty feet or more in height, and of strong and graceful proportions. The trunk divides at a slight angle into two or three arching limbs, and these again into many smaller curving and drooping branches. The trunk and the larger branches are often heavily fringed with short and leafy boughs. The tree is widely cultivated. Streets planted with it become columned and arched like the aisles of a Gothic cathedral. The wood is hard, and very tough from the interlacing of its fibers. It is used in making saddle-trees and for wheel-hubs, and is now largely exported to England to be used in boat- and ship-building. One day I found four men in a stone quarry, working with iron bars and rollers over a heavy flat slab. They were moving the stone slowly up a narrow plant into their cart. "John, " I said, "I would not think that board could hold a stone of such weight two minutes. Is it hickory?" "No sir, " said John, " that's an elm plank; it can't break." It did not break. It was one of the woods which the Deacon used in building his famous "one-hoss shay": So the deacon inquired of the village folk Where he could find the strongest oak, That count n't be split nor bent nor broke, - That was for spokes and floor and sills; He sent for lancewood to make the thills; The cross-bars were ash, from the straightest trees; The panels of whitewood, that cuts like cheese, But lasts like iron for thing like these; The hubs of logs from the Settler's Ellum; - Last of its timber, - they could n't sell 'em, Never an axe had seen their chips, And the wedges flew from between their lips, Their blunt ends frizzled like celery-tips;" --Oliver Wendell Holmes

Genus Ulmus, L. (Elm)

Leaves - simple; alternate; edge sharply and often doubly toothed. Outline - oval or egg-shaped, or…

Triangles are made of various substances such as wood, rubber, celluloid, and steel.

45 Degree Triangle

Triangles are made of various substances such as wood, rubber, celluloid, and steel.

Triangles are made of various substances such as wood, rubber, celluloid, and steel.

30-60 Degree Triangle

Triangles are made of various substances such as wood, rubber, celluloid, and steel.

Use a drawing board and T-square to test the accuracy of the 90 degree angle.

Testing Triangles

Use a drawing board and T-square to test the accuracy of the 90 degree angle.

Triangles are used in drawing lines at right angles to the T-square.

Drawing Vertical Parallel Lines

Triangles are used in drawing lines at right angles to the T-square.

It is possible to draw lines forming angles of 15 and 75 degrees by arranging two triangles.

Drawing Lines at an Angle 15 and 75

It is possible to draw lines forming angles of 15 and 75 degrees by arranging two triangles.

Weave pattern exercise: Divide A D and A B into 8 equal parts, and through the points O, P, Q, H, I, J, etc., draw horizontal and vertical lines. Now draw lines connecting O and H, P and I, Q and J, etc. As these lines form an angle of 45 degrees with the horizontal, a 45-degree triangle may be used. Similarly from each one of the given points on A B and A D, draw lines at an angle of 45 degrees to B C and D C respectively.

Weave Exercise

Weave pattern exercise: Divide A D and A B into 8 equal parts, and through the points O, P, Q, H, I,…

15 degrees with the horizontal or 75 degrees with the vertical.

Triangle Set Up for 15 Degrees

15 degrees with the horizontal or 75 degrees with the vertical.

30 degrees with the horizontal or 60 degrees with the vertical.

Triangle Set Up for 30 Degrees

30 degrees with the horizontal or 60 degrees with the vertical.

45 degrees with the horizontal or 45 degrees with the vertical.

Triangle Set Up for 45 Degrees

45 degrees with the horizontal or 45 degrees with the vertical.

60 degrees with the horizontal or 30 degrees with the vertical.

Triangle Set Up for 60 Degrees

60 degrees with the horizontal or 30 degrees with the vertical.

75 degrees with the horizontal or 15 degrees with the vertical.

Triangle Set Up for 75 Degrees

75 degrees with the horizontal or 15 degrees with the vertical.

Lines perpendicular to each other may be drawn by using a triangle in combination with the T-square. To draw a line perpendicular to a given line place a triangle against the T-square and move them together until the hypotenuse of the triangle matches the line. Turn the triangle on its right angled corner until the perpendicular line can be drawn on the hypotenuse of the triangle.

Drawing Perpendicular Lines Exercise

Lines perpendicular to each other may be drawn by using a triangle in combination with the T-square.…

Fahrenheit thermometer at 41 degrees. Blood heat, water freezes, and zero are all marked on the right side.

Thermometer

Fahrenheit thermometer at 41 degrees. Blood heat, water freezes, and zero are all marked on the right…

A machine that blasts hot air to produce temperatures from 600 to 900 degrees Fahrenheit.

Blast Furnace

A machine that blasts hot air to produce temperatures from 600 to 900 degrees Fahrenheit.

Sutures of the skull. Labels: a,a, the coronal suture, from the Latin corona, crown, so called from its situation on that part of the head, upon which the ancients placed the laurel, or olive crown, given to the victors in their games. It connects the frontal to the parietal bones; b, the sagittal suture, from a Latin word, signifying arrow, from its straight course. It runs from the middle of the frontal to the angle of the occipital bone, connecting the two parietals; c, the lambdoidal suture, extending from the sagittal suture down to the base of the brain on each side; e,e, the scaly overlapping of the temporal upon the parietal bones; hence called squamous suture.

Skull Sutures

Sutures of the skull. Labels: a,a, the coronal suture, from the Latin corona, crown, so called from…

Fourth rib. Labels: a, vertebral extremity, called the head, which is connected with the bodies of the two contiguous dorsal vertebrae. At b, the bone is contracted, forming the neck; c, is the tubercle at the back of the rib, which is articulated with the transverse process of the vertebrae; d, the angle; e, the sternal extremity; f, a groove for the intercostal vessels. This will serve for a general description of the ribs.

Fourth Rib

Fourth rib. Labels: a, vertebral extremity, called the head, which is connected with the bodies of the…

The sternum in this cut consists of two bones. The first is broad and thick above, and contracts as it descends. It is convex before and concave behind. At the upper angle a, the collarbone is articulated; b, the articular surface for the cartilage of the first rib; b, for the second rib; c,d,e,f,g, mark the articular surfaces of the 3rd, 4th, 5th, 6th, and 7th ribs; h, the ensiform cartilage, terminates the lower extremity of the sternum. In older people, this cartilages is often changed into bone.

Sternum

The sternum in this cut consists of two bones. The first is broad and thick above, and contracts as…

Scapula. Labels: a, superior angle; d, the glenoid cavity, or socket for the round head of the arm bone; m, the aeromion process; n, the caracoid process, which serve to protect the joint; f, the base; g, the costa, or inferior border, and h, the superior border of the triangle; l, the spine; o, the semilunar notch, for the passage of an artery, vein, and nerve.

Scapula

Scapula. Labels: a, superior angle; d, the glenoid cavity, or socket for the round head of the arm bone;…

A man exercising with chest weights. In this position, he extends his left arm out upward at a 45 degree angle.

Chest Weights

A man exercising with chest weights. In this position, he extends his left arm out upward at a 45 degree…

A diagram of semidiurnal and diurnal tide forces at various degrees of latitude.

Tidal Forces

A diagram of semidiurnal and diurnal tide forces at various degrees of latitude.

Three methods of expressing slopes: a, Slope Angle; b, Slope Ratio; c, Grade Percent.

Slope Methods

Three methods of expressing slopes: a, Slope Angle; b, Slope Ratio; c, Grade Percent.

The image shows surveying done using a slope board.

Slope Board

The image shows surveying done using a slope board.

The rectangular protractor used to measure angles by the military.

Rectangular Military Protractor

The rectangular protractor used to measure angles by the military.

The semicircular protractor used to measure angles by the military.

Semicircular Military Protractor

The semicircular protractor used to measure angles by the military.

This diagram shows the translation of contours to finding the measurement of the slope in grade percent, mils, degrees, and gradient.

Contours to Slope

This diagram shows the translation of contours to finding the measurement of the slope in grade percent,…

This figure "shows the cotidal lines and the lines of equal rise and fall for a diurnal component in latitude 30 degrees north." -Coast and Geodetic Survey, 1901

Cotidal Lines

This figure "shows the cotidal lines and the lines of equal rise and fall for a diurnal component in…

This figure shows "the cotidal lines and lines of equal amplitude for a diurnal tide in a circular sea of 20 degrees radius, the latitude of the center being 30 degrees north." -Coast and Geodetic Survey, 1901

Diurnal Cotidal Lines

This figure shows "the cotidal lines and lines of equal amplitude for a diurnal tide in a circular sea…

This figure shows "the cotidal lines and lines of equal amplitude for a diurnal tide in a circular sea of 20 degrees radius, the latitude of the center being 30 degrees north." -Coast and Geodetic Survey, 1901

Diurnal Cotidal Lines

This figure shows "the cotidal lines and lines of equal amplitude for a diurnal tide in a circular sea…

"It is seen that the motion of each particle is rectilinear and simple harmonic...The paths are vertical where...multiplies of 180 degrees and horizontal at points halfway between." -Coast and Geodetic Survey, 1901

Wave Motion

"It is seen that the motion of each particle is rectilinear and simple harmonic...The paths are vertical…

A man measuring the height of a tree by determining the angle and how far away he is standing.

Measuring Tree Height

A man measuring the height of a tree by determining the angle and how far away he is standing.

"The sun is so far away that it would appear at the same angle from Philadelphia, St. Louis, and Denver, and if it were noon at one of these places, it would be noon at the others. Vertical lines at the three places would be parallel. But when it is noon at St. Louis, it is 1 P.M. at Philadelphia and 11 A.M. at Denver." -Dryer, 1901

Earth Curvature

"The sun is so far away that it would appear at the same angle from Philadelphia, St. Louis, and Denver,…

An instrument used in geography to measure angles of the sun at different times of the day.

Heliotrope

An instrument used in geography to measure angles of the sun at different times of the day.

"BA is the ray of light passing through a rare medium (as, for instance, air); and upon its entrance into a denser medium (as, for instance, water) the ray will be deflected from the direction of its path BA, and will take the course AE. If the line CD is perpendicular to the dividing surface between the two media, then BAC is the angle of incidence and DAE is the angle of refraction." -Waldo, 1896

Atmospheric Optics

"BA is the ray of light passing through a rare medium (as, for instance, air); and upon its entrance…

Angles 1 and 2 are adjacent angles. Two angles with a common vertex and a common side between them are adjacent angles.

Adjacent Angles

Angles 1 and 2 are adjacent angles. Two angles with a common vertex and a common side between them are…

Angle 3 is an acute angle.

Acute Angle

Angle 3 is an acute angle.

Illustration of supplementary angles. Angles LMK and KMH are supplementary.

Supplementary Angles

Illustration of supplementary angles. Angles LMK and KMH are supplementary.

Illustration showing that the sum of angle 1 and angle 2 is angle ABC.

Sum of Angles

Illustration showing that the sum of angle 1 and angle 2 is angle ABC.

Illustration showing that the difference between angle 1 and angle 2 is angle ABC.

Difference of Angles

Illustration showing that the difference between angle 1 and angle 2 is angle ABC.

Illustration showing two positive angles; angle 1 being the acute angle and angle 2 being the reflex angle.

Acute and Reflex Angles

Illustration showing two positive angles; angle 1 being the acute angle and angle 2 being the reflex…

Illustration showing four angles that can be used to define different relationships, such as adjacent, supplementary, etc..

Relationships Between 4 Angles

Illustration showing four angles that can be used to define different relationships, such as adjacent,…

Illustration showing four angles that can be used to define different relationships, such as adjacent, supplementary, etc..

Relationships Between 4 Angles

Illustration showing four angles that can be used to define different relationships, such as adjacent,…

Illustration showing angles 1 and 2 are supplementary and angles ACD and DCB are supplementary. Also, Angles ACD and DCB are right angles.

Supplementary and Right Angles

Illustration showing angles 1 and 2 are supplementary and angles ACD and DCB are supplementary. Also,…

Illustration used to prove that all right angles are equal.

Equal Right Angles

Illustration used to prove that all right angles are equal.

Illustration showing that the sum of all the angles about a point equals 360°.

360° Sum of Angles

Illustration showing that the sum of all the angles about a point equals 360°.

Illustration showing that angles 1 and 2 are vertical and angles 3 and 4 are vertical.

Vertical Angles

Illustration showing that angles 1 and 2 are vertical and angles 3 and 4 are vertical.

Illustration showing that angles 1 and 2 are complementary.

Complementary Angles

Illustration showing that angles 1 and 2 are complementary.

Illustration showing that angles 1 and 2 are supplementary.

Supplementary Angles

Illustration showing that angles 1 and 2 are supplementary.

Illustration showing six angles that can be used to define different relationships, such as adjacent, supplementary, etc..

Relationships Between 6 Angles

Illustration showing six angles that can be used to define different relationships, such as adjacent,…

Illustration showing that if two angles of a triangle are equal, the bisectors of these angles are equal.

Angle Bisectors In An Isosceles Triangle

Illustration showing that if two angles of a triangle are equal, the bisectors of these angles are equal.

Illustration showing that if equal segments measured from the vertex are laid off on the arms of an isosceles triangle, the lines joining the ends of these segments to the opposite ends of the base will be equal.

Equal Segments In An Isosceles Triangle

Illustration showing that if equal segments measured from the vertex are laid off on the arms of an…

Illustration showing that if equal segments prolonged through the vertex are laid off on the arms of an isosceles triangle, the lines joining the ends of these segments to the opposite ends of the base will be equal.

Equal Segments In An Isosceles Triangle

Illustration showing that if equal segments prolonged through the vertex are laid off on the arms of…

Illustration used to show how to construct an angle equal to a given angle when given a vertex and a given side.

Construction Of An Equal Angle

Illustration used to show how to construct an angle equal to a given angle when given a vertex and a…

Illustration showing the three angle bisectors in a triangle.

Angle Bisectors In A Triangle

Illustration showing the three angle bisectors in a triangle.

Illustration showing how to construct the bisector of an angle.

Construction Of Angle Bisector

Illustration showing how to construct the bisector of an angle.

"Maximum extinction angles in the Pyroxene and Amphibole groups. Solid lines indicate extinction angles from c to c; broken lines from c to a. The extinction angle in an amphibole is generally less than 23 degrees; in a pyroxene it is generally greater. " -Johannsen, 1908

Extinction Angles

"Maximum extinction angles in the Pyroxene and Amphibole groups. Solid lines indicate extinction angles…