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Area of focus: Trigonometry and mathematical proofs

Topics covered:

> Trigonometry

> Right triangle definitions

> Trigonometric functions

> Special right triangles

> Law of sines

> Law of cosines

> Postulates and axioms

> Theorems

> Pythagorean Theorem

> Mathematical proof

Suggested time to complete (2 hrs):

> Teaching material (40 minutes)

> Practice activity (20 minutes)

> Final project (60 minutes)

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- 1. For the teacherʼs guide and all answersplease contact info@putterking.com Future KingsTM Education Program Subject: Math Level 3 (Queen)Putter King LLC All materials copyright © 2011Level 28 Shinagawa Intercity Tower A Putter King LLC2-15-1 Konan Minato-ku Tokyo, Japan All rights reserved 108-6028 Unit #: 2011010301
- 2. MathLevel 3 - Queen 2
- 3. SummaryArea of focus: trigonometry andmathematical proofsTopics covered: • Trigonometry • Right triangle deﬁnitions • Trigonometric functions • Special right triangles • Law of sines • Law of cosines • Postulates and axioms • Theorems • Pythagorean Theorem • Mathematical proofSuggested time to complete (2 hrs): • Teaching material (40 minutes) • Practice activity (20 minutes) • Final project (60 minutes) 3
- 4. TrigonometryTrigonometry is the study of the relations of the sides and angles of triangles. There are three special names given to triangles that tell how many sides (or angles) are equal. 60° 60° 60°Equilateral Triangle = = Scalene Triangle• Three equal sides • No equal sides• Three equal angles, always 60° • No equal angles Isosceles Triangle • Two equal sides • Two equal angles 4
- 5. Right Triangle DeﬁnitionsFor any right triangle that contains the angle A, the three sides of the triangle are: ▪ The hypotenuse is the side opposite the right angle: side c. The hypotenuse is always the longest side of a right triangle. ▪ The opposite side is the side opposite to the angle we are interested in: side a. ▪ The adjacent side is the side having both the angle of interest (angle A) and the right angle (angle C): side b. B c a (hypotenuse) (opposite) A b C (adjacent) 5
- 6. Trigonometric FunctionsTrigonometric functions are functions of an angle. They are used torelate the angles of a triangle to the lengths of the sides of a triangle. Function Abbreviation Description Sine sin opposite / hypotenuse Cosine cos adjacent / hypotenuse Tangent tan opposite / adjacent Cotangent cot adjacent / opposite Secant sec hypotenuse / adjacent Cosecant csc hypotenuse / opposite 6
- 7. Trigonometric FunctionsThere is a useful mnemonic device to help remember the trigonometric functions: SOH-CAH-TOA Sine = Opposite ÷ Hypotenuse Cosine = Adjacent ÷ Hypotenuse Tangent = Opposite ÷ Adjacent B a sin A = c b cos A = c a c (hypotenuse) (opposite) a tan A = b A b C (adjacent) 7
- 8. Special Right TrianglesA special right triangle is a right triangle whose sides are in a particular ratiowhich makes calculations on the triangle easier. Knowing the relationships ofthe angles or ratios of sides of these special right triangles allows one toquickly make calculations. 45º - 45º - 90º Triangles 30º - 60º - 90º Triangles 3 - 4 - 5 Triangles 5 2 1 4 2 3 1 3 1 8
- 9. Law of SinesThe law of sines is an equation relating the lengths of thesides of a plane triangle to the sines of its angles. a b c = = sin A sin B sin Cwhere a, b, and c are the lengths of the sides of a triangle, and A, B, and C are the opposite angles. b C a A B c 9
- 10. Law of CosinesThe law of cosines is an equation relating the lengths of thesides of a plane triangle to the cosine of one of its angles. c2 = a2 + b2 − 2ab cos C where a, b, and c are the lengths of the sides of a triangle, and A, B, and C are the opposite angles. b C a A B c 10
- 11. Postulates and Axioms Postulates are statements that are assumed to be true without proof. Postulates are also called axioms. Postulates serve two purposes: I) to explain undeﬁned terms II) to serve as a starting point for proving other statementsFor present day mathematicians, axiom and postulate hold a slightly different meaning than they did for the ancient Greeks. In the ﬁeldof mathematical logic, a clear distinction is made between two notions of axioms: logical (axioms) and non-logical (postulates). 11
- 12. TheoremsTheorems are statements that can be deduced and proved from deﬁnitions, postulates, and previously proved theorems. Famous theorems: • Pythagorean Theorem • Euclidʼs Proof of the Inﬁnitude of Primes • The Irrationality of the Square Root of 2 • sin2 θ + cos2 θ = 1 12
- 13. Pythagorean TheoremIn any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle). a2 + b2 = c2 b c a 13
- 14. Mathematical ProofA proof is a convincing demonstration that some mathematical statement is necessarilytrue. A proof must demonstrate that a statement is true in all cases, without a singleexception.A proof is a sequence of steps linked together by modus ponendo ponens (Latin for theway that afﬁrms by afﬁrming). This rule of logic says that if we know that “A implies B”,and if we know “A”, then we may conclude B. Typical Methods of Proof • Constructive Proof • Proof by Contrapositive • Proof by Contradiction • Proof by Induction • Counterexamples 14
- 15. Practice Activity #1 Find the length of: •p= •q= •r= Find the angle of: •θ= •β= 400 cm p β 300 cm 410.5 cm q 47.8º θ r 801.3 cm500 cm 500 cm 15
- 16. Practice Activity #2 a c b Arrange the large green pieces below to prove the Pythagorean Theorem using the proof by rearrangement method For full points, please give a written explanation of your proof.Hint: The miniature golf hole will be moreintuitive with the correct rearrangement.The gray pieces are golf course obstacles. 16
- 17. Final ProjectRequirementsDesign and label a miniature golf hole that includes at least: • special right triangles (2) • equilateral triangle (1) • isosceles triangle (1) • scalene triangle (1)For full points, be sure to: • use the Pythagorean theorem to ﬁnd the length of a side of a triangle in your design • use the law of sines to ﬁnd the length of a side or the measure of an angle of a triangle in your design • use the law of cosines to ﬁnd the length of a side of a triangle in your design • be creative! 17
- 18. Grading Rubric Bogey Par Birdie (70% - 79%) (80% - 89%) (90% - 100%) • Student did not include all • Student forgot to include a • Student included all required required triangles required triangle trianglesTriangles • Student incorrectly labeled the • Student incorrectly labeled • Student correctly labeled all triangles one triangle required triangles • Student did not use all of the required formulas to calculate • Student forgot to use one of • Student correctly used all the length of a side or the the required formulas required formulasFormulas measure of an angle • Student made a calculation • Student did not make any • Student made multiple error calculation errors calculation errors • Miniature golf hole design is • Miniature golf hole design is playable and aesthetically plain and simple • Miniature golf hole design is Design • Miniature golf hole design playable but rather plain pleasing • Miniature golf hole design is does not exhibit creativity creative and original 18

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