Understanding Engineering Mathematics

Solution to review question 5.1.2

A.(a) A full revolution is 360°, so a fraction (12

➤

)

p

q

of a full revolution

is

p

q

360 °, i.e.:

(i)^12 × 360 °= 180 ° (ii)^13 × 360 °= 120 ° (iii)^14 × 360 °= 90 °

(iv) 30° (v) 270° (vi) 225°

A full revolution is 2πradians, so a fraction

p

q

of a full revolution

is

p

q

× 2 π,i.e.

(i)

1

2

× 2 π=πradians (ii)

1

3

× 2 π=

2 π

3

radians

(iii)

1

4

× 2 π=

π

2

radians (iv)

π

6

radians

(v)

3 π

2

radians (vi)

5 π

4

radians

B. Referring to Figure 5.1aandcare bothsupplementary anglesto 40°

and soa=c= 140 °. b is thevertically oppositeangle to 40°and is

thus given byb= 40 °.

C. (i) By vertically opposite angles for intersecting lines we havea=

60 °in Figure 5.2(i). Then by corresponding anglesb=a= 60 °.

Finally, supplementary angles givec= 180 °− 60 °= 120 °.

(ii) Corresponding angles in Figure 5.2(ii) givea= 85 °,thenb= 85 °,

whence opposite angles givesc=b= 85 °. Corresponding angles

then givesd=c= 85 °. Supplementary angles finally givese=

180 °− 85 °= 95 °.

5.2.3 Triangles and their elementary properties

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144 160➤

Atriangleis any plane figure with three sides formed from line segments. There are three
types of interest – ascalene trianglehas three sides, and angles, of different lengths; an
isosceles trianglehas two sides of equal length standing on a base side with which the
two equal sides make equal angles; anequilateral trianglehas all three sides of equal
length and three equal angles (all 60°). A triangle in which all angles are acute is called
anacute-angled triangle. If one angle is 90°it is called aright-angled triangle. If one
angle exceeds 90°we say the triangle isobtuse.
The most basic property of a plane triangle is thatthe sum of its angles is 180°.(This
isnottrue for ‘spherical triangles’, drawn on the surface of a sphere – can you think of
a ‘triangle’ on a sphere that has 270°?) This property of plane triangles is worth proving
in terms of elementary facts of which we are already sure. We can do this for a general
triangleABC, acute-angle or obtuse, as follows. Draw a straight line, throughA, parallel