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35:The Sine Rule © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules

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The Sine Rule Triangles that aren’t Right Angled To find unknown sides and angles in non-right angled triangles we can use one or both of 2 rules: the sine rule the cosine rule The next few slides prove the sine rule. The cosine rule is on the next presentation. You do not need to learn the proof.

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The Sine Rule a, b and c are the sides opposite angles A, B and C A B C b a c ABC is a scalene triangle The Sine Rule

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A B C Draw the perpendicular, h, from C to BA. N h b a c In ABC is a scalene triangle

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The Sine Rule A h b a c b a c C B In ABC is a scalene triangle N

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The Sine Rule A B h b a c C In ABC is a scalene triangle In N

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The Sine Rule A BN h b a c A BN h b a c C In ABC is a scalene triangle

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The Sine Rule In ABC is a scalene triangle A BN h b a c A B h b a c C

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The Sine Rule so, and A B C ba c

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The Sine Rule A B C ba c so, and

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The Sine Rule... can be turned so that BC is the base. C A B ba c A B C b a c The triangle ABC... We would then get h

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The Sine Rule So, We now have So, and

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The Sine Rule The sine rule can be used in a triangle when we know One side and its opposite angle, plus One more side or angle Q qp Tip: We need one complete “pair” to use the sine rule. e.g. Suppose we know p, q and angle Q in triangle PQR The angle or side that we can find is the one that completes another pair.

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The Sine Rule B b a Solution: Use Ba b We don’t need the 3 rd part of the rule e.g. 1 In the triangle ABC, find the size of angles A and C. A B C 12 10 (3 s.f.)

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The Sine RuleSolution: As the unknown is a side, we “flip” the sine rule over. The unknown side is then at the “top”. Z y Y ( 3 s.f.) y ZY e.g. 2 In the triangle XYZ, find the length XY. Y Z X 13

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The Sine RuleSUMMARY The sine rule can be used in a triangle when we know One side and its opposite angle, plus One more side or angle If 2 sides and 1 angle are known we use: If 1 side and 2 angles are known we use: We write the sine rule so that the unknown angle or side is on the left of the equation a A

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The Sine RuleExercises 1.In triangle ABC, b = cm, c = cm and angle C =. Find the size of angles A and B. Solution: C c b 2. In triangle PQR, PQ = 23 cm, angle R = and angle P =. Find the size of side QR. B A C a

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The Sine Rule ( 3 s.f.) 2. In triangle PQR, PQ = 23 cm, angle R = and angle P =. Find the size of QR. r PR Solution: PQ R p Exercises

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The Sine Rule The following may be left out if time is an issue

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The Sine Rule e.g. In a triangle PQR, p = 5 cm, r = 7. 2 cm and angle P =. Drawing side r and angle P, we have: P Q 7.27.2 R1R1 5 This is one possible complete triangle. If an unknown angle is opposite the longest side, 2 triangles may be possible: one will have an angle greater than

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The Sine Rule This is the other. P Q R1R1 7.27.2 5 5 R2R2 e.g. In a triangle PQR, p = 5 cm, r = 7. 2 cm and angle P =.

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The Sine Rule This is the other. P Q R1R1 7.27.2 5 5 R2R2 The 2 possible values of R are connected since Triangle is isosceles e.g. In a triangle PQR, p = 5 cm, r = 7. 2 cm and angle P =.

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The Sine Rule This is the other. The 2 possible values of R are connected since Triangle is isosceles e.g. In a triangle PQR, p = 5 cm, r = 7. 2 cm and angle P =. P Q R1R1 7.27.2 5 5 R2R2 The calculator will give the acute angle ( < ). We subtract from to find the other possibility.

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The Sine Rule e.g. In a triangle PQR, p = 5 cm, r = 7. 2 cm and angle P =. Find 2 possible values of angle R and the corresponding values of angle Q. Give the answers correct to the nearest degree. Solution: P pr or

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The Sine Rule or: We have either: and P Q2Q2 7.27.2 5 R2R2 P Q1Q1 R1R1 7.27.2 5

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The Sine RuleSUMMARY If the sine rule is used to find the angle opposite the longest side of a triangle, 2 values may be possible. Use each value to find 2 possible values for the 3 rd angle. Use the sine rule and a calculator to find 1 value. This will be an acute angle ( less than ). Subtract from to find the other possibility.

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The Sine RuleExerciseSolution: B bc Find 2 possible values of angle ACB in triangle ABC if AB = 15 cm, AC = 12 cm and angle B =. Sketch the triangles obtained.

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The Sine Rule AB = 15 cm, AC = 12 cm, angle B = A B 15 C 12 A B 15 C 12 (i) (ii) Exercise

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The Sine Rule

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The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

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The Sine Rule a, b and c are the sides opposite angles A, B and C A B C b a c ABC is a scalene triangle The Sine Rule

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The sine rule can be used in a triangle when we know One side and its opposite angle, plus One more side or angle If 2 sides and 1 angle are known ( a, b and B ) we use: If 1 side and 2 angles are known ( A, b and B ) we use: We write the sine rule so that the unknown angle or side is on the left of the equation a A

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The Sine Rule We don’t need the 3 rd part of the rule B b a Solution: Use Ba b (3 s.f.) e.g. 1 In the triangle ABC, find the size of angles A and C. A B C 12 10

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The Sine Rule Solution: As the unknown is a side, we “flip” the sine rule over. The unknown side is then at the “top”. Z y Y ( 3 s.f.) y ZY e.g. 2 In the triangle XYZ, find the length XY. Y Z X 13

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The Sine Rule If an unknown angle is opposite the longest side, 2 triangles may be possible: one will have an angle greater than e.g. In a triangle PQR, p = 5 cm, r = 7. 2 cm and angle P =. Find 2 possible values of angle R and the corresponding values of angle Q. Give the answers correct to the nearest degree. Solution: P pr or

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We are now going to extend trigonometry beyond right angled triangles and use it to solve problems involving any triangle. 1.Sine Rule 2.Cosine Rule 3.Area.

We are now going to extend trigonometry beyond right angled triangles and use it to solve problems involving any triangle. 1.Sine Rule 2.Cosine Rule 3.Area.

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