GEOMETRY II AND FUNDAMENTALS OF TRIGONOMETRY
All of the shapes studied in Geometry I were bounded by straight lines. The areas of these figures could be calculated in a straight-forward manner by dividing the figures into parts with known areas and then adding the areas of each part. The perimeters were obtained by simple addition. There is one other plane figure remaining that is of great importance. This figure is the circle.
A circle is a plane closed curve such that every point on the curve is the same distance from a fixed point called the center of the circle. The length of the curve that forms the circle is called the circumference. (It could just as well be called a perimeter.)
The following figure is a circle. Point O is the center.
The distances OA, OB, OC, OD, and OE are equal.
The straight line that connects the center of the circle to any point on the boundary is called the radius. In the example above, each of the lines, OA, OB, OC, OD, and OE, is a radius. A straight line that connects opposite sides of the circle and passes through the center is called the diameter. There are clearly an infinite number of diameters in any circle, all of the same length. In the example above, the line DE is a diameter of the circle. The radius of a circle, or r, is half of the diameter, or D.
The radius of a circle intersects the circumference at a single point. When two different radii intersect the circumference, the length of the circumference between the two points of intersection is called an arc.
With reference to the figure above, we say that the radius subtends an arc of length AB.
In the very early days of the study of geometry, it was found that for any circle, the ratio of the circumference of the circle to the diameter of the circle was always constant. This ratio is given a special symbol, the Greek letter (pronounced pie). This ratio is true for any circle, regardless of its size:
The value of is 3.1415927. . . . . (where the dots indicate a continuing series of digits). When not using a calculator, using a value of = 3.1416 gives results that are sufficiently accurate. Since the diameter is twice the radius, we can express the circumference as C = 2r.
Find the circumference of a circle with a diameter of 3.6 inches.
When the areas of polygons were computed, it was found that they could be calculated by moving a line through some distance. This procedure is valid for a straight-sided figure, but it is not applicable to a circle. The area of a circle is given by the expression:
In terms of the diameter, the area is:
Find the area of a circle with a diameter of 3.6 inches.
The circle is special because it encloses a greater area for a given perimeter or circumference than any other plane figure.
In a study of plane geometry, the main concern is with the properties of two-dimensional figures; that is, their areas and perimeters. Now it is time to extend the concepts of plane geometry to solid geometry, or the study of objects that occupy space, or have volume. Solid volumes are generated by moving surfaces. For example, if a rectangle is moved in a direction perpendicular to its surface, a rectangular solid will be generated.
The rectangle shown on the front face is moved in a direction perpendicular to its surface. All of the angles of a rectangular solid are right angles, and a rectangular solid has six rectangular faces. The surface area of the solid is the sum of the surface areas of the six faces.
Volume is measured in terms of cubic units, such as cubic inches or cubic feet. In analogy to the unit of area, the unit cube is a rectangular solid in which each side has a length of 1 unit. The volume is then the number of these unit cubes that will completely fill the solid. The volume of a rectangular solid can be derived by examining the way in which it is generated. Starting with a rectangle of height h and width w, the area of the rectangle is A = hw. By moving this rectangle a distance 1 unit, it is in effect stacking up 1 of these rectangles 1 unit of distance apart. The volume is equal to the surface area of the rectangle times the distance through which this area has moved.
The surface area of the rectangular solid is the sum of the surface areas of the six faces. In the example above, the front and back faces each have an area of hw, the two side faces each have an area of lh, and the top and bottom faces have an area of lw. The total surface area is then:
Find the volume and surface area of a rectangular solid 4 inches long, 2 inches wide, and 3 inches high.
When each side of the rectangular solid is the same length, the figure is called a cube. A cube is generated by moving a square through a distance equal to the length of one of its sides. The volume and surface area of a cube can be found from the volume and surface area of a rectangular solid simply by making all the sides equal.
Notice that the volume of a rectangular solid is a length raised to the third power (cubed).
Take a rectangle and rotate it 360° about one of its edges. The resulting solid figure is called a cylinder.
The top and bottom faces of the cylinder are circles, and the straight line that connects the centers of the circles is the axis of the cylinder.
The height of the cylinder is the perpendicular distance between the two circular faces, or bases. The cylinder shown here is actually a special type of cylinder called a right circular cylinder. The axis of the cylinder is perpendicular to each base because the cylinder was generated by rotating a rectangle. If a parallelogram was used instead of a rectangle, a cylinder would still have been generated, but the axis would not be perpendicular to each base.
For now, the only concern will be with the right circular cylinder. If the width of the rectangle used to generate the cylinder is r, then each base will be a circle of radius r. If the length of the rectangle is h, then the height of the cylinder will also be h.
The volume of the cylinder is just the area of the circular base times the height of the cylinder. Again, this can be regarded as stacking h circles one unit of distance apart.
The lateral area of a cylinder is the area of the curved surface of the cylinder. This lateral area can be generated by moving the circumference of the circular base through the height of the cylinder. The circumference is 2r, and so the lateral area is 2rh. The total surface area of the cylinder is the lateral area plus the area of each base. Since each base is a circle, the total area of the bases is 2(r2). Thus, the total surface area of a cylinder is 2rh + 2r2.
Find the volume of a reinforcement rod that is 0.440 inches in diameter and 165 inches long.
Take a circle and rotate it 360° about one of its diameters and it will generate a solid figure known as a sphere. A sphere is a solid bounded by a curved surface such that every point on the surface is the same distance from a point within the solid. This point is the center of the sphere, and the distance is just the radius of the circle that generates the sphere. A diameter of the sphere is a straight line segment through the center with its ends on the surface. Its length is equal to twice the radius, as in a circle.
The volume of a sphere is not as easy to compute as the volume of a rectangular solid or cylinder. This is because the volume cannot be generated by moving a fixed area through some distance. The volume of the sphere is given by the expression:
Note that the volume is again in terms of the cube of a length.
What is the volume of a spherical tank 4 ft. in diameter?
The surface area of a sphere is given by the formula:
What is the surface area of a spherical pellet 0.42 inches in diameter?
A hemisphere is just half of a sphere, so its volume will also be half that of a sphere. Care must be taken, however, when calculating the surface area. If interest is only in the surface area of the curved surface, this area will be half the surface area of the sphere. If the total surface area is wanted though, it is necessary to include the area of the circular base.
The sphere has a unique property that distinguishes it from every other solid. For a fixed volume, the sphere has the smallest area of any solid. In other words, for the same surface area, we can enclose more volume with a sphere than with any other solid. This has application in the design of buildings, especially sports arenas, since a smaller surface area means both less material and less heat loss.
Trigonometry is the branch of mathematics that is concerned with the relationship between lines and angles. It is a very useful mathematical tool for science and engineering, for it provides techniques for indirect measurement. The height of a flagpole or the width of a river, for example, can be determined without climbing one or swimming the other. As will be shown, this is possible because the direct measurement of an angle allows us to calculate a length.
There are many ways of defining an angle. For our purposes, the most useful definition involves the rotation of a line about a point.
Angle A is the amount of rotation of line AB rotating about point A to line AC. AB is called the initial side, and AC is called the terminal side of the angle.
The size of angle A in the example above, in terms of amount of rotation, does not depend upon the lengths of line segments AB and AC. If these segments were extended indefinitely, the size of the angle would remain the same. The amount of rotation is measured in degrees, with 360° corresponding to one complete revolution of the line AC (terminal side). Notice also in the example above that line AC has been rotated in a counter-clockwise direction from the initial side. By convention, it is agreed that this shall be a positive angle. If the terminal side were rotated in the clockwise direction, it would result in a negative angle.
Notice that the same angle can now be described in two different ways. If the terminal side were rotated through a positive angle of 330°, it would be in the same position, relative to the initial side, as if it had been rotated through an angle of -30°.
The concept of a negative angle is consistent with the discussion of negative numbers in general because it indicates direction from some reference point. In the case of numbers, it shows direction opposite to the positive numbers on the number line, starting at a reference point called zero. In the case of angles, it shows rotation in the opposite direction from the reference point, or initial side.
Although the angle has been defined, its relationship to lengths has not been discussed. Suppose a rectangular, or x-y, coordinate system is drawn and an arbitrary point (x1, y1) in the first quadrant is chosen. Connect that point to the origin by a line called the radius vector. Angle A is the angle that the radius vector makes with the x-axis. Note that the radius vector is the terminal side of angle A, and that angle A is positive. The radius vector has length r1.
The x-coordinate is called the abscissa and the y-coordinate is called the ordinate. The abscissa, the ordinate, and the radius vector form a right triangle. Measure the ordinate and the radius vector and form their ratio:
Extend the radius vector so that it has a new length r2, but keep angle A the same. The radius vector will pass through a new point (x2, y2).
By measuring the new ordinate, y2, and the new radius vector, r2, and forming their ratio y2/r2, it is found that this ratio is exactly the same as the ratio y1/r1. For a fixed angle A, the ratio of the ordinate and the radius vector is constant. The ordinate and the radius vector can change, but their ratio does not. This ratio is given the special name sine (pronounced sign). Since this ratio depends only on the size of angle A, we say that it is a function of angle A, that is:
This is read as the sine of angle A is the ratio of the ordinate to the radius vector. (The word sine is abbreviated sin.)
The ordinate is nothing more than the projection of the radius vector onto the vertical, or y, axis. From the point (x, y) we construct a line perpendicular to the y-axis. The distance along the y-axis from the origin to the point of intersection is the projection of the radius vector.
The sine of an angle is a pure number since it is a ratio of two lengths. Associated with every angle, then, is a number called the sine of that angle that only depends on the size of the angle. This provides a tool for measuring an unknown length if by measuring one length and the angle, the other length can be calculated. It is often necessary to calculate the sine of every angle between 0° and 360°. To accomplish this, let us rotate the radius vector to see what happens to the ratio of the length of the y-axis projection and the radius vector. When angle A is 0°, the radius vector lies along the x-axis. Consequently, there is no projection along the y-axis, and so sin 0 = 0. As angle A increases, the projection along the y-axis also increases until A = 90°. At this point, the projection has the same length as the radius vector, so sin 90 = 1.
When the angle exceeds 90°, the projections begin to decrease until A = 180°. The radius vector now lies along the negative x-axis, and the projection along the y-axis is again 0°.
When the radius vector sweeps from 180° to 270°, the ratio of the projection on the y-axis to the radius vector again increases from 0 to 1, just as it does when the radius vector sweeps from 0° to 90°. In this case, however, the projection is along the negative y-axis, so the ratio is negative.
When angle A is 270°, the radius vector lies along the negative y-axis, so the ratio of the y-axis projection and the radius vector is -1 (the projection is along the negative y-axis).
Finally, when the radius vector sweeps from 270° to 360°, the y-axis projection decreases until, at 360° (or 0°), the projection is zero.
It is helpful to graph the variation of the sine of an angle as a function of the angle itself, to see how the sine changes as the angle changes from 0° to 360°. The sine of the angle is plotted on the vertical axis, and the angle is plotted on the horizontal axis.
There are a number of features of interest on this curve. Note that the absolute value of the sine of an angle can never be greater than 1. This is because the length of the projection of the radius vector onto the y-axis can never exceed the length of the radius vector itself. It can equal the length of the radius vector, as occurs at 90° and 270°, when the radius vector lies along the positive y-axis and the negative y-axis, respectively. At 0°, 180°, and 360° (or 0° again) the radius vector lies along the y-axis. The y-axis projection is 0°, so the sine of these angles is also zero. For any angle, then, the sine of the angle can be found by reading its value from the graph. In practice, these values are obtained more accurately from your calculator.
Notice also from the graph that two different angles may have the same sine. For example, the y-axis projection of the radius vector is the same length when the angle is either 45° or 135°, so the sine of these angles is the same:
The fact that two different angles may have the same sine will be of some importance when using the sine functions to solve problems involving triangles.
The sine function was the ratio of the projection of the radius vector onto the y-axis to the radius vector itself. Now project the radius vector onto the x-axis. As in the case of the sine function, the ratio of the x-axis projection to the radius vector depends only on the angle. The projection is called the abscissa, and the ratio of the abscissa to the radius vector is called the cosine of the angle A.
The word cosine usually is abbreviated cos. The cosine function can be evaluated as the radius vector sweeps from 0° to 360° in exactly the same manner as for the sine function. For example, when angle A is 0°, the radius vector lies along the x-axis, meaning the radius vectors projection onto the x-axis has the same length as the radius vector. Therefore, cos 0 = 1. As the radius vector goes from 0° to 90°, the x-axis projection, or abscissa, decreases.
When angle A is 90°, there is no projection on the x-axis, so cos 90 = 0. Just as for the sine function, a graph of the cosine function can be constructed:
This graph is very similar to that of the sine function. The maximum absolute value of the cosine is one, since the length of the abscissa can never exceed that of the radius vector. In this case, the maximum values occur at 0° and 180° where the radius vector lies along the positive x-axis or negative x-axis, respectively. At 90° and 270°, the value of the cosine is 0 since the radius vector lies along the y-axis and hence has no projection onto the x-axis. By using this graph, the cosine of any angle between 0° and 360° can be found.
The sine and cosine functions have been defined in terms of the projection of the radius vector onto the y- and x-axis, but these definitions can be made much more general. The radius vector, the projection on the y-axis (the ordinate), and the projection on the x-axis (the abscissa) form the three sides of a right triangle:
With regard to angle A, the sine of A is the ratio of the ordinate to the radius vector. It is also the ratio of the side opposite the angle to the hypotenuse:
Similarly, the cosine of A is the ratio of the side adjacent the angle to the hypotenuse:
These definitions apply to each of the two acute angles of the right triangle. For example, compute the sine and cosine of each acute angle in the right triangle shown.
This example illustrates an important point. Notice that the sine of angle A is equal to the cosine of angle B. Also, note that angle A plus angle B is equal to 90°, since the two acute angles of a right triangle must always total 90°. Therefore, A = 90 - B, and sin A = sin (90 - B) = cos B. By referring back to the graphs of the sine and cosine functions, it can be seen that by sliding the sine curve 90° to the left, it becomes a cosine curve (or by sliding the cosine curve 90° to the right, it becomes a sine curve). Two angles that add up to 90° are called complementary angles. Therefore, the sine of an angle is always equal to the cosine of the complementary angle, and vice versa.
There is one more very important trigonometric ratio to be studied, namely, the tangent.
Compute the tangent of each acute angle:
Unlike the sine or cosine, the tangent of an angle can have a value greater than 1. Notice that the tangent of an angle is just the reciprocal of the tangent of the complementary angle:
These three functions, the sine, cosine, and tangent, are the foundations of trigonometry. Knowing an angle and a side, these functions can be used to determine the unknown sides of a right triangle. For any angle, the sine, cosine, or tangent can be obtained from your calculator.
There are three other trigonometric ratios that are related to the ratios that have already been presented. They are included here for the sake of completeness, although they are not needed for most engineering calculations. The cosecant, abbreviated csc, of an angle is just the reciprocal of the sine of the angle:
The secant, abbreviated sec, is the reciprocal of the cosine:
The cotangent is the reciprocal of the tangent:
These three functions do not introduce anything new, but they are convenient for some problems. Remember that they are just reciprocals of the three primary functions, and so can be determined from them.
In summary, the six trigonometric functions are described as follows:
The methods developed so far enable us to answer questions such as "what is the sine of 30°?" or "what is the cosine of 77°?" There are occasions, however, when we need to know the opposite question; that is, "what is the angle whose sine is 0.5?" This is given by the arc function, or inverse function, and has the same relationship to the trigonometric functions as does the antilogarithm to the logarithmic function. The angle whose sine is 0.5 would be indicated as: arc sin 0.5 or sin-1 0.5
The term arc can be used as a prefix to any of the trigonometric functions. Using the inverse notation sin-1, it is important to realize that the -1 is not an exponent. It merely indicates that the inverse is to be found. Your calculator can be used to compute these functions.
In studying equations, it is found that all equations are of two types: conditional equations or identities. An identity is an equation that is always true regardless of the value of the unknown. There are relationships between the trigonometric functions that are always true, regardless of the size of the angle. These relationships are called trigonometric identities.
Some of these identities have previously been presented. For example, the sine function is defined as:
The cosecant function is:
Therefore, for any angle A, it must be true that:
Another identity involves the sine, cosine, and tangent. The sine function is:
The cosine function is:
Dividing the sine by the cosine, the result is:
However, this is just the tangent function. Thus, for any angle A:
One of the most important trigonometric identities can be developed using the Pythagorean Rule. As an example, the right triangle shown below is to be evaluated.
It is known that: a2 + b2 = c2.
Dividing this equation by c2 results in:
However, is just sin A, and is cos A. Therefore: sin2 A + cos2 A = 1
As a final example, the use of the Pythagorean Rule again (but this time dividing the equation by b2), results in:
is the tangent of A, and is the secant of A.
Therefore for any angle A: tan2 A + 1 = sec2 A.
There are many trigonometric identities, but these are the most common. The value of these identities lies in the fact that they frequently permit the simplification of a problem. For instance, an expression like sin2 37.3 + cos2 37.3 can be immediately replaced it by the value 1.
The trigonometric functions are all double-valued; that is, two different angles have the same sine, cosine, or tangent. This fact can be used to relate the trigonometric function of any angle greater than 90 to the trigonometric function of an angle between 0 and 90. For any angle between 90 and 180. sin = sin (180 - ) cos = -cos (180 - ) tan = -tan (180 - )
For example: sin 150 = sin (180 - 150) = sin 30 cos 150 = -cos (180 - 150) = -cos 30 tan 150 = -tan (180 - 150) = -tan 30
When the angle lies between 180 and 270: sin = -sin ( - 180) cos = -cos ( - 180) tan = tan ( - 180)
For example: sin 240 = -sin(240 - 180) = -sin 60
Finally, when the angle lies between 270 and 360: sin = -sin (360 - ) cos = cos (360 - ) tan = -tan (360 - )
Angles normally are measured in degrees, with 360° defining a complete revolution. There is another measure of angle, however, that frequently arises in science and engineering. This unit is the radian, abbreviated rad. The radian is defined with respect to a circle. Suppose that a circle of radius, r, is rotated such that the radius subtends an arc of length, s, on the circumference of the circle.
In rotating, the radius has moved through an angle called . When the length of the arc is exactly equal to the radius of the circle, the angle is defined to be 1 radian:
When s = r, = 1 radian.
A relationship between radians and degrees can be found by rotating the radius through one complete revolution, or 360°. The length of the arc will then be the circumference of the circle, 2r. How many times must an arc of distance r be added up so that it equals the circumference of a circle? The answer is 2 times. Therefore, 360° is divided into 2 segments, and by definition each segment is equal to 1 radian. Therefore:
Radians can be converted to degrees by multiplying by:
Change three radians to degrees:
To convert degrees to radians, simply perform the inverse operation, i.e., multiply by:
Change 90° to radians.
Since 360° corresponds to 2 radians, these conversions will present no problem.