SHS2 Additional Mathematics · Semester 2, Week 4
Measurement of Triangles
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Curriculum details
- Strand
- Geometric Reasoning and Measurement (Strand 2)
- Sub-strand
- Measurement of Triangles (2.2)
- Content standard
- 2.2.2.CS.1 - Demonstrate understanding of trigonometric identities and apply algebraic techniques to verify identities and solve Trigonometric problems on them. 2.2.2.LO.1 Find trigonometric values using compound, multiple and half angles and prove the sine and cosine rule. 2.2.2.LO.2 Verify whether or not a given trigonometric equation is an identity, and solve trigonometric equations.
- Indicator
- 2.2.2.LI.1 - Prove and apply compound angles to derive the identities for multiple angles and half angles.
- Suggested placement
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Semester 2, Week 4
(Week 24 of the year)
Our suggestion, laid out in curriculum order across three terms of twelve weeks. NaCCA does not fix the week, so follow your school's scheme of learning.
- Source document note
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The official curriculum document has a fault in this entry, so the curriculum text above is transcribed exactly as printed:
- exemplars - p.365: adjacent duplicated CambriaMath characters were collapsed, but this PDF also maps some equation glyphs to the wrong letter or operator; verify every mathematical expression in this row against the rendered page
- exemplars - p.365: a stacked fraction, matrix, vector or other two-dimensional construct is flattened by the text layer; all visible parts require comparison with the rendered page
- Curriculum reference
- NaCCA curriculum document, p. 365
Exemplars (from the NaCCA curriculum)
Project-based Learning, Collaborative Learning, Talk for Learning Learning Experience: Learners in mixed-ability groups research how to derive compound angles from basic trigonometric ratios and share their findings in class. Activity 1: Proofs of compound angles Group leader shares their findings on compound angles with the class as follows: 1. (𝐴 + 𝐵) = 𝑠 + 𝑐 - 𝑠 𝑠 (𝐴 − 𝐵) = 𝑠 − 𝑐 - 𝑐 𝑐 (𝐴 + 𝐵) = 𝑐 − 𝑠 - 𝑐 𝑐 (𝐴 − 𝐵) = 𝑐 + 𝑠 G&%R$G&%T - 𝑡 𝑡 (𝐴 + 𝐵) = !*G&%RG&%T - 𝑡 𝑡 (𝐴 − 𝐵) = G&%R*G&%T !$G&%RG&%T Activity 2: Application of identities of compound angles Learners in groups state the importance of the identities in Activity 1 and apply their knowledge of special angles to solve practical examples without using calculators. Example Express 𝑐150° in a surd form. Solution: 𝑐150° = 𝑐 (90 + 60)° 𝑐 (𝐴 + 𝐵) = 𝑐 − 𝑠 𝑐(90 + 60)° = 𝑐 90°𝑐 60°- 𝑠 90°𝑠 60° 1 √3 = 0 £ ¤ − (1) u v 2 2 √3 𝑐150° = − 2 Learners in their mixed-ability group create questions on compound angles for other groups to solve and present their solutions to the class. Activity 3: Proofs of Multiple angles Learners in their mixed-ability research on multiple angles and group leaders share their findings in class. Condition If two angles are such that 𝐴 = 𝐵 then - 𝑠2𝐴 = 2𝑠 Or 𝑠2𝐵 = 2𝑠 - 𝑐2𝐴 = 𝑐 ) 𝐴 − 𝑠 ) 𝐴 )G&%R - 𝑡2𝐴 = !*G&% ! R Example: Given that 𝐵 = 30°, find 𝑠2𝐴 if angle 𝐴 is equal to angle𝐵. Solution: 𝑠2𝐴 = 2𝑠30°𝑐30° 1 √3 √3 =2 × = 2 2 2 - Learners in their mixed-ability groups create questions on multiple angles for other groups to solve and present their solutions to the class. Activity 4: Proofs of half angles Learners in their mixed-ability groups research on half angles, and group leaders share their findings in class. R Condition: If A is an angle, then half of A is represented by ) and hence Half angle formula for the sine function 𝐴 1 − 𝑐 𝑠 = ±ë 2 2 Half angle formula for the cosine function 𝐴 1 + 𝑐 𝑐 = ±ë 2 2 Half angle formula for the tangent function 𝑡 = ±y R !*;POR OL%R !*;POR or !$;POR or ) !$;POR OL%R Example: Find the value of 𝑠15° using the half-angle formula. Solution: 𝐴 = 15° ⟹ 𝐴 = 30° 2 30° 1 − 𝑐30° 𝑠 = ±ë 2 2 𝑠15° = ±0.2588 Learners, in their mixed-ability groups, create questions on half angles for other groups to solve and present their solutions to the class. Teaching and Learning Resources: - Graph paper - Ruler - Scientific calculator - Protractor - Computer Assessment (2.2.2.AS.1). The document marks these depth-of-knowledge levels for this indicator: Level 2 Skills of conceptual understanding; Level 4 Extended critical thinking and reasoning.