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

ISBN 0-471-15045-2

**Download**(direct link)

**:**

**35**> 36 37 38 39 40 41 .. 126 >> Next

Example 2. Is the relation P = {(2,1) (3, 2) (2,3) (1, 5)} a function?

Solution. No, because one member of the domain of {2, 3,2, 1} is repeated in the domain, which is 2.

If a relation is given in the form of a graph, we can apply the “vertical line test” to see if it is a function. In figure g, if a vertical line, which is shown dashed, is drawn anywhere on the graph and it intersects the graph in at most one point, then the graph is of a function. On the other hand, if the straight line intersects the graph in more than one point, then the graph is not of a function. In the examples in figure g, graph A is of a function and graph B is not of a function.

References: Arrow Graph, Cartesian Coordinates, Domain, Equations, Graphs, Ordered pairs, Range.

CORRESPONDING ANGLES

Reference: Alternate Angles.

CORRESPONDING SIDES

Suppose a triangle ABC is enlarged to obtain its image A'B'C'. The image of AB is A!B' and they are a pair of corresponding sides. The corresponding sides AB and A!B' are opposite equal angles A and A'. The other two pairs of corresponding sides are AC and A'C\ and BC and B'C1.

&

y\

R

(g)

122 COSINE RULE

Corresponding sides occur whenever one polygon is an enlargement of another, and we say the polygons are similar.

References: Congruent Figures, Enlargement, Similar Figures.

COSINE

Cosine is usually abbreviated cos.

Reference: Trigonometry.

COSINE RULE

The cosine rule is a set of trigonometric formulas connecting the three sides of a triangle with the cosine of one of the angles of the triangle. It is mainly used in triangles that are not right-angled, because simpler methods are used for right-angled triangles, as explained under the entry Trigonometry. The notation used for the cosine rule, and also for the sine rule, is that the sizes of the three angles of the triangle are referred to using capital letters, say A, B, and C, and the lengths of the sides are lower case letters, say a, b, and c. The convention is that side a is opposite to angle A, and similarly for sides b and c.

A

(a)

The cosine rule states that in any triangle

a2 = b2 + c2 — 2be cos A

and this rule is used for finding the length of side a.

There are two more variations of the same rule:

b2 = a2 + c2 — 2a c cos B This is used for finding the length of side b c2 = a2 + b2 — 2a b cos C This is used for finding the length of side c

COSINE RULE

123

Each of these three forms of the cosine rale can be rearranged in order to use them to calculate the angles of a triangle. The rearranged forms are

b2 + c2 - a2 cos A =---------—---------------- For finding angle A

a2 + c2 — b2 cos B =-------------------------- For finding angle B

2 ac

a2 + b2 - c2

cos C = For finding angle C

2 ab

It is important at this stage to recognize which type of triangle can be solved using the cosine rale and which type of triangle requires the sine rale. The rules explaining which to use are listed here:

1. There is no need to use these two rules for right-angled triangles, because it is easier to use sine, cosine, or tangent.

A

(b)

2. If the lengths of two sides and the angle between the two sides are given in the problem, then use the cosine rale to find the third side b. In the example in figure a, the angle of 47° lies between the two sides of 6 cm and 4 cm, and therefore the cosine rule is used to find the length of side b.

3. If the lengths of three sides of the triangle are given, use the rearranged form of the cosine rale to find one of the angles.

4. For all other triangles use the sine rale.

Example 1. On her bedroom wall Amanda has hung a small picture, but it does not hang straight (see figure b). The lengths of the strings are AC = 8.4 cm AB = 7.3 cm, and the angle BAC = 84°. Use the cosine rule to find the length of BC.

124

COSINE RULE

A

8.4 cm C

(o)

Solution. Write down what we know about the triangle ABC:

b = 8.4 cm, c = 7.3 cm, A = 84°

We are given two sides and the angle between them, and are to find the length of the third side, which is a. Since we are finding the length of the side of a triangle, there are three versions of the cosine rale from which to choose. We use the following to find a: a2 = b2 + c2 — 2be cos A. Write

a2 = 8.42 + 7.32 - 2 x 8.4 x 7.3 x cos 84°

a2 =70.56 + 53.29- 12.82

a2 = 111.03 a = Vi 11.03 a = 10.54 (to 2 dp)

The length of BC is 10.54 cm.

Take care with your calculator when the angle you are working with is obtuse. Say the angle A is obtuse; then the expression 2be cos A is negative, because the cosine of an obtuse angle is negative. The next example demonstrates how to find the size of an angle using the rearranged form of the cosine rale.

Example 2. The line QR in figure c represents a grassy bank 3 meters long; a ladder QP, which is 4.1 meters long, rests against a brick wall and reaches 2 meters up the wall. Find the size of angle PRQ. There is a peg at the point Q to stop the ladder slipping down the grassy bank.

Solution. Draw the triangle PQR, but rename it ABC, because the cosine rule refers to a triangle ABC. Write down what we know about triangle ABC: a = 3, b = 2,

**35**> 36 37 38 39 40 41 .. 126 >> Next