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

ISBN 0-471-15045-2

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(a)

Now that we have established the names of the sides of the right-angled triangle in relation to a given angle, we can define the three trigonometric ratios sine, cosine, and tangent. Suppose we draw a right-angled triangle and let one of the acute angles be 6Â° (see figure b). The lengths of the sides are labeled O, A, and H in the usual way. This triangle is now enlarged to produce additional, similar right-angled triangles with sides of lengths 01? A1? and Hi, and so on. All the triangles are similar; they have a right angle and have an angle of 6Â°. One of the properties of these triangles is that the ratios O/A are all equal to the same value, say k. This can be expressed as

O _0i_02_0,_k

A Ai A2 A3

458 TRIGONOMETRY

Os

(b)

The value k is defined to be the tangent of the acute angle 9. Similarly, the ratios

0 01 O2 $3

H = JT1 = Jh = lh

are all equal to the sine of the angle 6, and the ratios

A A\ A2 A3

H = H~1 = H~2 = H~3

are all equal to the cosine of the angle 9. The abbreviation for sine is sin, that for cosine is cos, and that for tangent is tan.

These definitions are used to solve problems regarding right-angled triangles, and are summarized here (see figure c):

sin#

0

H*

COS#

A

H*

tan#

In the following examples ensure that your calculator is in degree mode, which will be indicated on the calculator screen as DEG.

Example 1. A ladder of length 3 meters is placed against a vertical wall and makes an angle of 70Â° with the horizontal ground (see figure d). Calculate how far up the wall the ladder reaches.

Solution. Figure d is a sketch of the problem, with the distance required marked as x. Write

0 = x 0 is opposite to the angle of 0 =70Â°

H = 3 H is opposite the right angle

TRIGONOMETRY

459

(d)

Select the trigonometric ratio that uses O and H, which is the sine:

sin 70Â° = |

O

3 x sin 70Â° = JC

Multiplying both sides of the equation by 3

x =2.82 (to 2 dp) Using the calculator

The ladder reaches 2.82 meters up the wall.

Example 2. A road has a gradient of 1 in 7 (see figure e). Find the angle the road makes with the horizontal.

Solution. A gradient of 1 in 7 means that the road rises 1 meter vertically for every 7 meters in a horizontal direction. Suppose the road makes an angle of 6Â° with the horizontal; this is the angle we have to find. Write

0 = 1 O is opposite to the angle of 6

A =1 A is adjacent to the angle of 6

Select the trigonometric ratio that uses O and A, which is the tangent:

i o

tan 9 = \ tan# = â€”

1 A

6 = tan-1 (i) If tan 6 = a, then tan-1 a = 6, provided 6 is

an acute angle

6 = 8.1Â° (to 1 dp) Using the calculator

The road makes an angle of 8.1Â° with the horizontal.

Example 3. The length of a shadow cast by a tree is 24.5 meters. If the sunâ€™s rays are shining at an angle of 67Â° with the horizontal, calculate the height of the tree (see figure f).

9

7

(e)

460 TRIGONOMETRY

Solution. The angle of 67Â° can be described as the angle of elevation of the sun. The right-angled triangle has its sides designated as O, A, H. Write

O = h O is opposite to the angle of 47Â°

A = 24.5 m A is adjacent to the angle of 47Â°

(f)

Select the trigonometric ratio that uses O and A. Write

h O

tan 47 =................ tan 0 =

24.5 A

h = 24.5 x tan 47Â° Multiplying both sides by 24.5

h = 26.3 (to 1 dp) Using the calculator

The height of the tree is 26.3 meters.

Example 4. Figure g shows the end view of a bam. The height of the roof is 3 meters and the length of the roofing is 6 meters. Find (a) the pitch of the roof, which is the angle shaded in the figure, and (b) the width of the bam, which is the distance BC.

(9)

TRIGONOMETRY 461

Solution. Using the symmetry of the figure, we can see that angle CAD is half the shaded angle. The triangle ACD is right-angled, and we first find angle CAD.

(a) Write (see figure h)

H = 6 H is opposite to the right angle A = 3 A is adjacent to angle CAD

Select the trigonometric ratio that uses H and A, which is the cosine:

(h)

cos (angle CAD)

COS0

H

Angle CAD = cos 11 - | If cos 9 = a, then cos 1 a = 9, provided 9 is

\^/ an flpulp anrrlf^

Angle CAD = 60Â°

an acute angle Using the calculator

Since the shaded angle is twice angle CAD, the pitch of the roof is 120Â°.

(b) The width of the bam is twice the length of CD. In triangle ADC

Angle CAD = 60Â° Found in part (a)

H = 6 H is opposite the right angle

O = CD O is opposite the angle of 60Â°

Select the trigonometric ratio that uses H and O, which is the sine:

CD O

sin 60Â°

sin 9

H

6 x sin 60Â° = CD Multiplying both sides of equation by 6

CD = 5.196 (to 3 dp) Using the calculator

Therefore

Width of bam = 10.392 meters Width = 2 x CD

462 TURNING POINTS

References: Acute Angle, Angle of Elevation, Enlargement, Gradient, Horizontal, Pythagorasâ€™ Theorem, Ratio, Right Angle, Similar Figures, Vertical.

TRINOMIAL

A trinomial is an algebraic expression that contains three terms. Examples of trinomials are

x + 2a â€” b, 3 ay â€” y4 + 2, 2x2 â€” 3x + 5

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