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MADAM MUNIRAH BINTI SHAMSUDDIN KOLEJ KEMAHIRAN TINGGI MARA,  LENGGONG

SITI MAISARAH BINTI MOHD AZIZI LG 2002755

OBLIQUE TRIANGLES (COSINE RULE) &  AREA  OF TRIANGLES

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This video lesson will show you how you can use the   Law of Cosines ,  a  formula  to help you find the missing side or angle of any kind of triangle.

In this video lesson, we are specifically looking at  Oblique Triangles   to use the  Law of  Cosines  and to  finding the  Area of  Triangles.

Then, we will explore the steps to finding the  Area of Triangles.

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Any kind of triangle that is not a right triangle is an  Oblique Triangle .

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In any triangle, the square of any side is equal to the sum of the squares of the other two sides minus twice the product of these two sides multiplied by the cosine of the included angle.

a 2  = b 2  + c  2  –  2bc.cos  A b 2  = a 2  + c  2  –  2ac.cos  B c 2  = a 2  + b  2  –  2ab.cos  C

cos  A  =  b 2  +  c 2   – a 2                    2bc cos   B  =  a 2  +  c 2  –  b 2                     2ac cos   C  =  a 2  +  b 2  –  c 2                     2ab

In reference to the triangle shown in Figure A, the   Law of Cosines , for any triangle  ABC  is :

The  Law of Cosine  is also known as  Cosine Rule .

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T wo sides and one included angle is given.

Cosine Rule  can be used when the following values are given:

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Figure 1 shows the triangle ABC. Given  that ∡B =  68 ° , AB  = 20.5 cm,  BC  = 12.2 cm.  Find  the length of AC.

Example 1:  Given 2 sides and included angle.

b2
= (20.5)2 + (12.2)2 - 680
= 408.04 + 148.84 + 187.38
= 369.5
= 19.22 cm

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Figure 2 shows the triangle PQR.   Given that ∡R =  111.4 ° ,  PR = 20 cm, RQ = 25 cm. Find :  a) Length PQ  b) ∡ Q

Example 2:  Given 2 sides and included angle.

a)
b)
r2
sin Q
20cm
sin Q
= 202 + 252 - cos 111.40
= 400 + 625 + 364.87
= 1389.87
= 37.28 cm
sin 111.4
37.92cm
37.9cm
= 0.4913
= 29.430

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Figure 2 shows the triangle ABC.   Given  that AB = 6.4 cm, BC = 4.7 cm and AC = 7.8 cm. Find :  a) ∡A  b) ∡C

sin LC
b)
6.4
sin
sin
sin37.20
4.7
6.4sin37.020
4.7
= 0.8199
= 55.070

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Now let’s continue our lesson to finding  Area  of Triangle …

How to use  Law of Cosine  or  Cosine Rules ?

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What is  Law of Cosine  or  Cosine Rules ?

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Area of Triangle  is an any space that is inside the borders of a triangle shape.

The measurement or size of a surface on a shape is called the  area .

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In reference to the  Area of  Triangle  shown in Figure C, the formula for finding the  Area of Triangle  is:

1
ax bx sin C
2
1
bxcx sin A
2
1
ax c x sin B
2

To find the  area  of a shape, we need to use a formula.

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In Figure  4  shows Δ PQR. Such that PQ = 13 cm, QR = 10 cm and  ∡ Q =  57.8 ° .  Find the area of the triangle Δ PQR.

The area of the triangle PQR
1
x 13 x 10 sin 57.80
2
= 55.00 cm2

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In figure 5   shows Δ ABC such that AC = 14 cm, BC = 11 cm and  ∡ C =  125 ° .  Find:  AB  ∡ B  c) The area of the triangle ABC

a)
b)
c)
A 82
= (14)2+
= 196 + 121 + 176.7
A 82
sin ZB
14
sin XB =
cos 1250
= 22.22 cm
sin 1250
22.22
14xsin1250
22.22
= 31.10
Area of the triangle ABC
1
x 14 x 11 x sin 1250
2
= 63.1 cm2
= 0.5161

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Using the Cosine Rule, solve  triangle PQR in which p = 6.5 cm, q = 7.4 cm and ∠R = 58 °. What is the approximate length of side r ?

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Use the information given in the figure to determine the value of c to the nearest tenth of a meter.  a = 12.3 ,  b =  17.6.

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Solve the triangle ABC.   Given  that AB = 6.4 cm, BC = 4.7 cm and  AC  = 7.8 cm. Find  ∡A.

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In  Δ ABC  given  that AC = 14 cm, BC = 11 cm and  ∡ C =  125 ° .  Find:  AB  ∡ B

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Which a = 27.2 cm,  b  = 26.3 cm, and  ∠ C = 92.2 °.  Find the  area  of a  triangle ABC.

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Which j = 6.9 cm,  k  = 9.7 cm, l = 14 cm and  ∠ L = 113.93 °.  What is the area of this triangle JKL ?

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Enquiry:   sitimaisarah320.dpd@lenggong.kktm.edu.my

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