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C = it x 3 Substituting D = 3 in the formula C = ixD
C =9.42 (to 2 dp) Using it in the calculator
The circumference of the pond is 9.42 meters.
A = 7t x 1.52 Substituting R = 1.5 into the formula A = ttR2,
where D -f 2 = R
A = 7.07 (to 2 dp) Using it in the calculator The area of the pond is 7.07 square meters.
References: Area, Center of a Circle, Changing the Subject of a Formula, Chord, Diameter, Graphs, Perimeter, Radius, Secant, Sector of a Circle, Segment of a Circle, Semicircle, Tangent.
76 CIRCULAR FUNCTIONS
CIRCLE GEOMETRY THEOREMS
A list of the theorems is given here. They are explained in detail under their separate entries.
♦ The angle at the center of a circle is twice the angle at the circumference. See Angles at the Center and Circumference of a Circle.
♦ The angle in a semicircle is a right angle. See Angle in a Semicircle.
♦ Angle in the Alternate Segment.
♦ Angles on the same arc are equal. See Angles on the Same Arc.
♦ Converse of angle in a semicircle theorem. See Angle in a Semicircle.
♦ Converse of angles on the same arc are equal. See Cyclic Quadrilateral.
♦ Converse of opposite angles of a cyclic quadrilateral = 180°. See Cyclic Quadrilateral.
♦ The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle. See Cyclic Quadrilateral.
♦ Radius is perpendicular to the tangent. See Tangent and Radius Theorem.
♦ The sum of opposite angles of a cyclic quadrilateral = 180°. See Cyclic Quadrilateral.
♦ Tangents from a common point are equal. See Tangents from a Common Point.
♦ Intersecting chords theorems. See Intersecting Chords.
Reference: Geometry Theorems.
This is another name for trigonometric functions. Three circular functions with their abbreviations in brackets are sine (sin), cosine (cos), and tangent (tan). They are defined in two ways:
1. An elementary definition, which involves a right-angled triangle, which is given in the entry Trigonometry.
2. A more comprehensive definition, given here, which involves a circle, hence the name circular functions. The definitions of sin, cos, and tan relate to an angle of 6 degrees, which must be carefully explained.
The circle in figure a has a radius of 1 unit, and is called the unit circle. The origin of the axes is 0, the center of the circle. P(x, y) is any point on the circle, and the length OP is 1 unit. Angle 6 is measured counterclockwise from the positive direction of the x-axis. As the radius OP rotates about the point 0, the angle 6 increases in size.
Figure b shows that 6 increases from 0° to 90° to 180° to 270°. After a full turn the angle 6 = 360°, which is at the same place as 6 = 0°. For convenience the four sectors
CIRCULAR FUNCTIONS 77
of the circle are named the first, second, third, and fourth quadrants, respectively, as shown in figure b.
Using the unit circle, we can now state the definitions of the three circular functions in terms of the coordinates of the point P :
cos 9 = x, sin# = y, tan# = —, providedx is not equal to zero
The circular functions sin 6, cos 6, and tan 9 can take positive and negative values, depending on the values of x and y in each of the quadrants of the circle.
♦ First quadrant: When 9 is between 0° and 90°, both the x and y coordinates are positive, so both cos 9 and sin 9 are positive, and so is tan 9.
♦ Second quadrant: When 9 is between 90° and 180°, x is negative and y is positive, so cos 9 is negative and sin 9 is positive, and tan 9 is negative.
♦ Third quadrant: When 9 is between 180° and 270°, x is negative and y is negative, so cos 9 is negative and sin 9 is negative, and tan 9 is positive.
♦ Fourth quadrant: When 9 is between 270° and360°, x is positive and y is negative, so cos 9 is positive and sin 9 is negative, and tan 9 is negative.
The same cycle is repeated for angles from 360° to 720°.
The graphs of the three circular functions are sketched in figure c, but only for values of 9 from 0° to 360°, they can be extended horizontally for all real values of 9. The tan graph has asymptotes at 9 = 90°, 270°, 450°,..., because the tan of each of these angles is undefined since x = 0 at these angles. These asymptotes are repeated every 180°.
References: Asymptote, Trigonometry.
This is the center of the circle that passes through the three vertices of a triangle, or vertices of any polygon. This circle is called the circumcircle of the triangle ABC (see figure a). This circumcenter can be outside the triangle if it has an obtuse angle. A circumcircle can always be drawn through the vertices of any triangle, but this is not necessarily true for other polygons.
Example. Figure b is a map of a park that has an arts center (A), a museum (M), and the park gates (G). The town council has decided to build a toilet block (T) (see figure c), which is to be equal distances from G, A, and M. Find where to build the toilet block.