The Cosine Rule

Keywords

[table][br][br][tr][br][td]Cosine Rule[/td][br][td]코사인 법칙[/td][br][td]余弦定理[/td][br][td]余弦定理[/td][br][/tr][br][tr][br][td]Pythagorean Theorem[/td][br][td]피타고라스 정리[/td][br][td]피타고라스의 정리[/td][br][td]勾股定理[/td][br][/tr][br][tr][br][td]Triangle types[/td][br][td]삼각형 유형[/td][br][td]삼각형의 유형[/td][br][td]三角形类型[/td][br][/tr][br][tr][br][td]Sides and angles relationship[/td][br][td]변과 각의 관계[/td][br][td]변과 각의 관계[/td][br][td]边与角的关系[/td][br][/tr][br][tr][br][td]Angle determination[/td][br][td]각도 결정[/td][br][td]각도 결정[/td][br][td]角度确定[/td][br][/tr][br][tr][br][td]Sine Rule[/td][br][td]사인 법칙[/td][br][td]사인 법칙[/td][br][td]正弦定理[/td][br][/tr][br][/table]

Inquiry questions

[table][br][tr][br][td][b]Factual Inquiry Questions[/b][/td][br][td][b]Conceptual Inquiry Questions[/b][/td][br][td][b]Debatable Inquiry Questions[/b][/td][br][/tr][br][tr][br][td]What is the formula of the Cosine Rule and in which type of triangle can it be applied?[/td][br][td]Why does the Cosine Rule work for any type of triangle, and what does it tell us about the relationship between the sides and angles of a triangle?[br][/td][br][td]Is the Cosine Rule more valuable in practical applications than the Sine Rule?[/td][br][/tr][br][tr][br][td]How does the Cosine Rule relate to the Pythagorean Theorem?[/td][br][td][table][tr][td][br]How can the Cosine Rule be used to determine the size of an angle given the lengths of all three sides of a triangle?[/td][/tr][/table][br][/td][br][td]How does the understanding of the Cosine Rule impact the development of technology, especially in fields requiring precise measurements and calculations?[/td][br][/tr][br][tr][br][td][br][/td][td][/td][br][td]Could the principles of the Cosine Rule be applied to non-Euclidean geometries, and what would be the implications?[/td][br][/tr][br][/table][br][br]

Trigonometry Cosine Rule - find a side

Cosine rule - Finding a missing angle

Checking for understanding

Question 1[br]In triangle ABC, side a = 7 cm, side b = 24 cm, and side c = 25 cm. What is the measure of angle C?[br][br]

B) [math]60^{\circ}[/math][br] D) [math]30^{\circ}[/math] C) [math]45^{\circ}[/math][br] A) [math]90^{\circ}[/math][br]

Question 2[br]Using the cosine rule, how can you find the length of a side in a triangle if you know two sides and the angle between them?[br][br][br]

C) [math]c=\sqrt{a^2+b^2+2ab\cos(C)}[/math][br] D) [math]c=a^2-b^2-2ab\cos(C)[/math][br] B) [math]c=a^2+b^2-2ab\cos(C)[/math][br] A) [math]c=\sqrt{a^2+b^2-2ab\cos(C)}[/math][br]

Question 3[br]In a triangle with sides of lengths [math]a=8cm[/math], [math]b=15cm[/math], and the angle [math]C=60^{\circ}[/math]between them, what is the length of side [math]c[/math]?[br][br][br]

A) 23 cm B) 19 cm D) 17 cm C) 21 cm

Question 4[br]If in a triangle, side a = 5 cm, side b = 12 cm, and the angle C = 90^\circ\ between them, what is the length of side c ?[br]

C) 13 cm[br] A) 10 cm[br] D) 14 cm B) 11 cm[br]

Question 5[br]Which statement is true regarding the application of the cosine rule?

C) It applies to any triangle to relate the sides and angles.[br] D) It provides the relationship between the sides and angles of a square.[br] A) It can only be used in right-angled triangles.[br] B) It is used to find the lengths of sides in polygons.[br]

[b]Exam style questions involving cosine rule[/b][br]Question 6, 11, 12, 14,[br][br]Question 40. leads you to proving the cosine rule.

[MAA 3.1-3.3] 3D GEOMETRY - TRIANGLES

[MAA 3.1-3.3] 3D GEOMETRY - TRIANGLES_solutions

Optional extension

Proof of cosine rule will not be examined but it can be wonderful to be able to prove the theorems.