Constitución del campo científico de los estudios sociales acerca del desarrollo

If the three sides of a triangle are known but no angle, then an alternative equation can be used to determine the area of the triangle. Let us take the area of a triangle (e.g. that in Figure 10.32) as being given by ½bc sin A. If we square both sides of the equation we obtain:

(area)² = (½bc sin A) x (½bc sin a) = ¼b²c² sin² A

But sin2A + cos2A = 1 (see Chapter 11). Thus we can write (area)2 = ¼b2c2(l - cos2 A)

We can write this as:

(area)2 = ¼b2c2(1 - cos A)(1 + cos A)

Using the cosine law for triangles we can write for cos A:

cos A = b2 + c2 - a2

2bc Hence:

(area)² = ¼b²c²

(

) (

)

= ¼b2c2

(

) (

)

= 1

16 [a2 - (b - c)2] [(b + c)2 - a2]

= 1 16 (a - b + c)(a + b - c)(b + c + a)(b + c - a)

Let a + b + c = 2s, i.e. s is half the length of the perimeter of the triangle.

Then subtracting 2b from both sides of this equation gives:

a - b + c = 2(s - b)

Subtracting 2a from a + b + c = 2s gives:

b + c - 2a = 2(s - a) and if we had subtracted 2c:

a + b - c = 2(s - c) Thus we obtain:

(area)2 = ¹

16 x 2(s - b) x 2(s - c) x 2s x 2(s - a) area = √ s(s - a)(s - b)(s - c)

Example

Determine the area of a triangle which has sides of a = 6 m, b = 5 m and c = 9 m.

We have 2s = a + b + c = 6 + 5 + 9 = 20. Hence s = 10. Thus:

area = √ 10(10 - 6)(10 - 5)(10 - 9) = √ 10 x 4 x 5 x 1 = √ 200 Hence the area is 14.1 m².

Revision

33 Determine the areas of the following triangles:

(a) Sides a = 3 m, b = 4 m, c = 5m, (b) Sides a = 6 m, b = 8 m, c = 10 m, (c) Sides a = 7 m, b = 10 m, c = 12 m, (d) Sides a = 5 cm, b = 7 cm, c = 8 cm, (e) Sides a = 7 cm, b = 8 cm, c = 9 cm, (f) Sides a = 5 cm, b = 6 cm, c = 7 cm, (g) Sides a = 4 cm, b = 5 cm, c = 7 cm.

34 Determine the area of a parallelogram which has adjacent sides of lengths 2.4 cm and 3.2 cm and the length of the shortest diagonal is 2.0 cm.

Problems 1 Determine, to three decimal places, the values of the following:

(a) sin 23°, (b) cos 15.1°, (c) tan 19.5°, (d) cosec 34°, (e) sec 59°, (f) cot 57°, (g) sin 0.50, (h) cos 1.21, (i) tan 0.56, (j) cosec 0.85, (k) sec 1.05, (1) cot 1.23, (m) sin 140°, (n) cos 150°, (o) tan 130°, (p) sin 190°, (q) cos 195°, (r) tan 230°, (s) sin 280°, (t) cos 290°, (u) tan 315°, (v) sin 320°, (w) cos 350°, (x) sin 261.2°, (y) sec 301.8°, (z) cot 342.3°.

2 Determine using a calculator the values between 0° and 90°, to one decimal place, of the following angles:

(a) The angle which has a sine of 0.78, (b) The angle which has a cosine of 0.15, (c) The angle which has a tangent of 1.9.

3 Solve for x, to two decimal places, in the following right-angled triangles:

(a) Adjacent 10, opposite x, angle 68°, (b) Adjacent 4, hypotenuse x, angle 55°, (c) Opposite 6, adjacent x, angle 23°, (d) Adjacent 12, opposite x, angle 58°, (e) Hypotenuse 4, adjacent x, angle 70°, (f) Hypotenuse 2, opposite x, angle 38°.

4 What is the height of a flagpole if the angle of elevation of the top of the pole from a point on the level ground 10 m from its base is 70°?

5 A ladder of length 6 m leans against a vertical wall with its base on horizontal ground. If the ladder makes an angle of 70° with the ground, how far is the bottom end from the wall?

6 The roof of a house is as shown in Figure 10.36. What is the length AB if AC = 9.0 m?

7 A and B are two towns 30 km apart. If B is on a bearing 50° east of north from A, how far east is B from A?

8 A guy wire of length 5 m to a pole rising from level ground makes an angle of 62° with the ground. How high above the ground is the wire attached to the pole?

9 In Figure 10.37, AB represents the jib of a crane and BC the tie. If AB is 10.0 m and BC is 4.0 m, what is the length of the tie?

10 Observer A sees an aeroplane as directly overhead. Observer B, who is 300 m due east of A, sees the plane as being at an angle of elevation of 70°. What is the height of the plane?

11 A ramp has to be built with a slope of 15° to enable a heavy machine to be slid up it to a height of 1.3 m. What must be the length of the ramp needed to achieve this?

12 A roof has to span 12 m with one slope pitched at 40° to the horizontal and the other at 50°. What will be the height of the ridge above the eaves?

13 Show that tan 60° - tan 30°

1 + tan 60° tan 30° = 1

√3 .

14 Determine the angles between 0° and 360°, to one decimal place, having the trigonometric ratios:

(a) sin θ = 0.35, (b) cos θ = -0.55, (c) cos θ = 0.60, (d) tan θ = 0.10, (e) tan θ = -1.5, (f) sin θ = -0.48, (g) cot θ = -1.24, (h) sec θ = - 2.5, (i) cosec θ = -2.5, (j) cosec θ = 1.5.

15 Determine, to two decimal places, the values of the following:

(a) sin ^π/6, (b) tan π/2, (c) cos ^3π/2, (d) sin 5π/6, (e) tan 7π/4, (f) cos 77π/6, (g) tan 77π/6.

16 Determine, using the sine rule, the required sides and angles for each of the following triangles:

(a) Angle A = 68°, angle C = 70°, side c = 42 mm, determine angle B, sides a and b,

(b) Angle A = 58°, side a = 180 mm, side b = 150 mm, determine angles B and C, side c,

(c) Angle C = 25°, angle B = 80°, side b = 10 cm, determine angle A and sides a and c,

(d) Angle = 10°, angle C = 60°, side b = 50 mm, determine angle B and sides a and c,

(e) Angle B = 50°, side b = 30 mm, side c = 20 mm, determine angles A and C and side a,

(f) Angle A = 70°, angle C = 40°, side c = 100 mm, determine angle B and sides a and b,

(g) Angle A = 62°, angle C = 62°, side b = 50 mm, determine angle B and sides a and c,

(h) Angle A = 53°, angle B = 62°, side a = 125 mm, determine angle C and sides b and c,

(i) Angle A = 80°, side a = 165 mm, side b = 130 mm, determine angles B and C and side c,

(j) Angle A = 30°, side a = 50 mm, side b = 80 mm, determine angles B and C and side c.

17 A roof has a span of 9 m with eaves that slope on one side at 44° to the horizontal and on the other side at 36° to the horizontal. Determine the lengths of the sloping parts of the roof.

18 A wall is leaning outwards at an angle to the vertical of 15°. A ladder of length 2 m is placed against the wall with its base on level ground 1 m from the base of the wall. How high up the wall does the ladder reach?

19 A vertical pole of length 6 m stands alongside a road which slopes away from it with a constant slope. The pole casts a shadow which is 14 m long directly downhill when the angle of elevation of the sun is 56°.

What is the angle of the road with the horizontal?

20 Determine, using the cosine rule, the required sides and angles for each of the following triangles:

(a) Side a = 12, side b = 63, side c = 52, determine angles A, B and C, (b) Angle A = 57.5°, side b = 78, side c = 97, determine side a and angles B and C,

(c) Side a = 9, side b = 13, side c = 17, determine angles A, B and C, (d) Side a = 7, side b = 5, side c = 3, determine angles A, B and C, (e) Side a = 4, side b = 7, side c = 5, determine angles A, B and C, (f) Angle A = 80°, side b = 7, side c = 8, determine side a and angles B and C,

(g) Angle A = 66°, side b = 40 mm, side c = 50 mm, determine angles B and C and side a,

(h) Side a = 40 mm, side b = 50 mm, side c = 70 mm, determine angles A, B and C,

(i) Angle C = 74°, side a = 100 mm, side b = 50 mm, determine angles A and B and side c.

21 A plane flies for 50 km in a straight line from its start point before turning through 20° and flying in a straight line on the new bearing for 70 km. What distance is the plane from the start?

22 A parallelogram has adjacent sides of lengths 55 mm and 82 mm with an angle between them of 68°. What is the length of the diagonal through this corner of the parallelogram?

23 A parallelogram has adjacent sides of lengths 30 mm and 50 mm with an angle between them of 144°. What is the length of the diagonal through this corner of the parallelogram?

24 Determine the areas of each of the following triangles:

(a) Angle A = 85°, side b = 40 mm, side c = 10 mm, (b) Angle B = 130°, side a = 40 mm, side c = 60 mm, (c) Angle C = 52°, side a = 2 m, side c = 5 m, (d) Angle A = 55°, side b = 12 m, side c = 4 m, (e) Side a = 4 cm, side b = 7 cm, side c = 9 cm, (f) Side a = 2 m, side b = 2 m, side c = 2 m, (g) Side a = 5 cm, side b = 12 cm, side c = 13 cm.

25 A parallelogram of area 30 cm2 has adjacent sides of lengths 6 cm and 9 cm. Determine the angles of the parallelogram.

26 A parallelogram has adjacent sides of lengths 9 cm and 5 cm. If the shortest diagonal has a length of 7 cm, what is the area of the parallelogram?

11.1 Introduction This chapter extends the trigonometry in Chapter 10 by considering the use of trigonometric identities in the solution of problems where we have such expressions as sin² A, cos² A, sin (A + B), cos (A + B), sin 2A and cos 2A.

Such types of expressions are frequently encountered in engineering and often need to be converted into other forms to aid the manipulation of equations.

11.2 Trigonometric