## Sine And Cosine Rule

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This is a mathematical presentation which enables you to determine the sides and angles of a particular triangle when the triangle itself is not a right angled triangle.

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HE SINE RULE
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The Sine Rule is used to solve any problems involving triangles when at least either of the following is known: a) two angles and a side b) two sides and an angle opposite a given side In Triangle ABC, we use the convention that a is the side opposite angle A b is the side opposite angle B The sine rules enables us to calculate sides and angles In the some triangles where there is not a right angle.
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Example 2 (Given two sides and an included angle) Solve triangle ABC in which ZA = 550, b = 2.4cm and c = 2.9cm By cosine rule, a2 = 2.42 + 2.92-2 x 2.9 x 2.4 cos 550 = 6.1858 a = 2.49cm sinB sin550 2.4 2. 487 2.4 sin55a 2.487 = 52.20 zc = 1800 - 550 - 52.20 72.80
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Using this label of a triangle, the sine rule can be stated Either sin A sin B sin A sin B Or Use [1] when finding a side Use [2] when finding an angle sin C sin C [2]
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Example: 7cm Given Angle ABC Angle ACB = 500 Find c. To find c use the following proportion: sin C sin 500 sin B 22-11=? sin600 7 xsin50 sin 60 6.19 (3 s.F)
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In ABAC AC = 6cm, BC = 15cm and Z A Find ZB SOLUTION: sin B sin B sin A sin 1200 6 x sin 600 B = 0.346 20.30 = 1200 6 cm 15 cm 1 200 11 11 11
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{9 O DRILL: SOLVE THE FOLLOWING USING THE SINE RULE: Problem 1 (Given two angles and a side) In triangle ABC, ZA = 590, LB = 390 and a = 6.73cm. Find angle C, sides b and c. Problem 2 (Given two sides and an acute angle) In triangle ABC, ZA = 550, b = 16.3cm and a = 14.3cm. Find angle B, angle C and side c. Problem 3 (Given two sides and an obtuse angle) In triangle ABC ZA =1000, b = 5cm and a = 7.7cm Find the unknown angles and side.
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Answer Problem 1 LC = 180 。 - ( 39 。 十 59 。 ) P 9 b 6 . 7 3 引 n 3 9 。 引 n 5 9 • : • 6 , 7 3 : : + : : : 引 n 3 9 。 引 n 5 9 ' : ' 4 , 94c ㄇ ( 3.s ) 6 . 7 3 C s i ㄇ 8 2 。 引 n 5 9 。 C 引 n 5 9 。 = 7.78 ( 3s. ㄏ . )
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ANSWER PROBLEM 2 14.3 sin 550 16.3 sin B 16.3 sin 690 C sin 560 16.3 sin 550 sinB= 14.3 = 0.9337 LB = 69.00 zc=1800 -690 -550 = 560 16.3 sin 560 sin 69 0 = 14.5 cm (3 SF)
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Answer Problem 3 引 n 1 0 0 。 S i n B 5 5 引 n 1 0 00 P 9 = O. 6 3 9 5 引 1 0 . 6395 LB ZC = 180 。 - ( 39 . 8 。 + 1 000) 4 〔 I. 2 。 引 n 1 0 0" s i n 4 〔 I. 2 。 7 , 7 引 n 4 〔 I. 2 。 C 引 n 1 0 0 。 = 5 .05c ㄇ ( 3s )
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∕ … 一 「 Q 9
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Sometimes the sine rule is not enough t s Ive for a non-right angled triangle. {9 F r example: 18 300 In the triangle shown, we do not have enough inforn to use the sine rule. That is, the sine rule only provi Following: a sin 300 14 sin B 18 sin C Where there are too many unknowns.
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For this reason we derive another useful result, known as the COSINE RULE. The Cosine Rule maybe used when: a. Two sides and an included angle are given. b. Three sides are given The cosine Rule: To find the length of a side a2 = b2+ c2 - 2bc cos A b2 = a2+ c2 - 2ac cos B c2 = a2 + b2 - 2ab cos C
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HE COSINE RULE: To find an angle when given all three ides. cos A = 2bc cos B = 2ac 223 cos C= 2ab
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{9 Example 1 (Given three sides) In triangle ABC, a = 4cm, b = 5cm and c = 7cm. Find the size of the largest angle. The largest angle is the one facing the longest side, which is angle c. casc = 2 2 2 x ax b 2 2 2 2 -0.2 LC = cas 1 101 .50 0.2 (Id.p. )
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I Be very sure to keep these formulas in your mind
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