# The A to Z of mathematics a basic guide - Sidebotham T.H.

ISBN 0-471-15045-2

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References: Cycle, Frequency.

ANALYTICAL GEOMETRY

Reference: Cartesian Coordinates.

ANGLE

This is a measure of turning or rotation; the units of angle measurement are degrees or radians. The angle x in the figure is the amount of turning between two rays OA and OB. The symbol for angle is /. The angle in the figure can be represented by the three capital letters, IAOB, or by a single small letter, Lx. It is also common in trigonometry to represent an angle by the letters of the Greek alphabet, for example, 19, La, or If.

22 ANGLE BETWEEN A LINE AND A PLANE

o

B

References: Acute Angle, Degree, Radian, Ray.

ANGLE BETWEEN A LINE AND A PLANE

This topic is part of three-dimensional trigonometry. To find the angle between a line and a plane, the line is projected onto the plane, and then the angle between the projected line and the original line is calculated. This angle is the angle between the line and the plane.

Suppose a straight nail ON is hammered into a piece of wood at an angle so that the nail is not upright (see figure a). The projection of the nail ON onto the plane of the wood is OW, as shown in figure a. The projection of ON onto the plane can be considered to be the shadow cast by the nail ON when parallel rays of light shine at right angles to the plane.

The angle NOW is the angle between the line ON and the plane of the wood. The method of calculating the angle between a line and a plane is explained in the following example.

Example. The longest diagonal of a cuboid, which is a box, is the line DF (see figure b). Calculate the angle between the line DF and the base EFGH.

(b)

(c)

ANGLE BETWEEN TWO PLANES 23

Solution. The projection of the line DF onto the base is the line HF. So the angle the line DF makes with the base EFGH is the angle DFH.

In the process of finding this angle, we use two right-angled triangles, HEF and DHF. The first step is to calculate the length of HF, using triangle HEF, which is positioned in the base of the cuboid (figure c):

HF2 = HE2 + EF2 Pythagoras’ Theorem HF2 = 22 + 42 Substituting HE = 2 meters and EF = 4 meters

HF2 =4+16 Squaring 2 and 4

HF = V20 Taking the square root

In triangle DHF, using V20 as the length of HF gives

Using tangent of an angle opposite side adjacent side

If tan a = b, then a = tan-1 b Using calculator

The angle between the line DF and the plane base EFGH is 33.9°.

References: Angle between Two Planes, Plane, Pythagoras’ Theorem, Trigonometry.

Tangent (angle DFH )

V2Ö

Angle DFH = tan 1 Angle DFH= 33.9° toi dp

ANGLE BETWEEN TWO PLANES

This topic is part of three-dimensional trigonometry. Suppose two planes which are inclined to each other intersect in the straight line XY (see figure a). The two planes can be thought of as hinged together, rather like an opening trapdoor. If the sloping plane swings down onto the other plane, then the angle through which it turns is the angle between the two planes. This angle is also called the dihedral angle with respect to the two planes.

To find the angle between the two planes, we select two lines, one in each plane, which intersect at the hinge XY (see figure b). Both lines are at right angles to

24 ANGLE BETWEEN TWO PLANES

XY. The angle between these two lines is the angle between the two planes. This concept is demonstrated in the following example, in which one of the planes is a triangle.

Example. Figure c shows an upright, square-based pyramid. Calculate the angle between the plane ABCD, which is the base of the pyramid, and the sloping plane VBC.

Solution. The two lines we select in the planes are VM and OM, as both lines are at right angles to their line of intersection BC. These lines are selected because they are two sides of a right-angled triangle VOM, whose lengths we can find. The required angle between the two planes is angle VMO.

In triangle VOM, the length of OM is 2 cm, and the length of VM is found from the right-angled triangle VMB in the following way (see figure d):

B o

VM2 + MB2 = VB2 Pythagoras’ Theorem

VM2 + 22 = 52 Substituting lengths MB = 2 and VB = 5

VM2 =25 — 4 Squaring 5 and 2, and rearranging equation

VM = -/25 — 4 Taking square roots

VM = Subtracting the numbers

ANGLE BISECTOR 25

The length of VM is now used in triangle VOM (see figure e):

Using cosine of an angle

adjacent side

hypotenuse

Angle VMO = cos 1

If cos a = b, then a = cos 1 b

Angle VMO = 64.1° to 1 dp Using calculator

The angle between the plane ABCD and the sloping plane VBC is 64.1°.

References: Angle, Angle between a Line and a Plane, Dihedral Angle, Pyramid, Pythagoras’ Theorem, Square Root, Trigonometry.

ANGLE BISECTOR

This is a line that divides an angle into two equal angles. The angle bisector has a raler-and-compass construction, which is described in the example below.

Example. A piece of pizza has been left for Jacob and Nathan’s supper. Doing a drawing of the pizza, show how to construct a line, using ruler and compasses only, that equally divides the piece of pizza into two.

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