Measurement of Triangles · SHS2 Additional…

Lesson notes

Learning Objectives

Indicator: 2.2.2.LI.1 - Prove and apply compound angles to derive the identities for multiple angles and half angles.

By the end of the lesson, learners can:

Prove the compound angle identities for sine, cosine and tangent for both (A + B) and (A − B) using basic trigonometric ratios.
Apply the compound angle identities to find exact trigonometric values of angles such as 15°, 75° and 150° in surd form without using a calculator.
Derive the multiple angle identities for sin 2A, cos 2A and tan 2A from the compound angle identities by setting A = B.
Derive the half angle identities for sin(A/2), cos(A/2) and tan(A/2) from the multiple angle identities.
Use the half angle identities to find exact trigonometric values, such as sin 15°, using special angles.

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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: 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: The official curriculum document has a fault in this entry, so the curriculum text above is transcribed exactly as printed:

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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.