Measuring Triangles

Curriculum details

Strand

Geometric Reasoning and Measurement (Strand 2)

Sub-strand

Measuring Triangles (2.2)

Content standard

3.2.2.CS.2 - Demonstrate understanding of compound angles and multiple angles and solve related problems 3.2.2.LO.1 Draw and analyse basic trigonometric graphs using values of the unit circle, maximum values and minimum values.

Indicator

3.2.2.LI.2 - Use knowledge in compound angles to derive harmonic identities and find the maximum and minimum values of a trigonometric function.

Suggested placement

Semester 2, Week 1 (Week 21 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:

• exemplars - p.503: 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

Curriculum reference

NaCCA curriculum document, p. 503

Exemplars (from the NaCCA curriculum)

Talk for Learning, Group work, and Building on what others say.
Learning Experience: Learners in mixed-ability investigate the maximum and minimum values.
Activity 1: Expression of 𝑎 + 𝑏 as 𝑅(𝜃 − 𝛼) Learners in mixed-ability groups brainstorm and learn how to express 𝑎 + 𝑏 as 𝑅(𝜃 − 𝛼)
Activity 2:
- Learners in groups use the addition formula for cosine to obtain the expansion of 𝑅(𝜃 − 𝛼) to obtain (𝑥) 𝑐 𝑐 (𝛼) + 𝑅 𝑠 𝑠 (𝑥) 𝑠 𝑠 (𝛼)
- Learners brainstorm to find 𝑅 and 𝛼 by comparing and equating expressions on both sides of the equations, that is
- 𝑅 𝑐 𝑐 (𝛼) and 𝑅 𝑠 𝑠 (𝛼) = 𝑏
- ∴ 𝑎 ) + 𝑏 ) = 𝑅 ) (𝛼) + 𝑅 ) (𝛼) , Factorise 𝑅 ) to obtain 𝑅 ) [(𝛼) + (𝛼) ] = 𝑎 ) + 𝑏 ) , since [(𝛼) + (𝛼) = 1] ' 𝑅 = √𝑎 ) + 𝑏 ) , 𝑅 > 0 and 𝑡 𝑡 (𝛼) = &
Note: 0 ≤ 𝜃 ≤ 90°
Example 1: Express 4 𝑐 𝑐 (𝑥) + 3 𝑠 𝑠 (𝑥) in the form 𝑅 𝑐 𝑐 (𝑥 − 𝛼) .
Solution: Let 𝑅 𝑐 𝑐 (𝑥 + 𝛼) = 5 𝑐 𝑐 (𝑥) − 12 𝑠 𝑠 (𝑥) ⇒ (𝑥) 𝑐 𝑐 (𝛼) − 𝑅 𝑠 𝑠 (𝑥) 𝑠 𝑠 (𝛼) = 5 𝑐 𝑐 (𝑥) − 12 𝑠 𝑠 (𝑥) By comparison, 𝑅 𝑐 𝑐 (𝛼) = 5 and 𝑅 𝑠 𝑠 (𝛼) = 12 𝑅 = √5 ) + 12 ) = 13 𝑢 !) 𝑡 𝑡 (𝛼) = .
!) ⇒ 𝛼 = i . j = 67.4°
∴ 5 𝑐 𝑐 (𝑥) − 12 𝑠 𝑠 (𝑥) = 13 𝑐 𝑐 (𝑥 + 67.4°)
Activity 3: Learners in mixed-ability groups brainstorm how to identify the maximum or minimum value. Using Example 1, learners identify 13 as the maximum value.
Teaching and Learning Resources:
- Scientific Calculator
Assessment (3.2.2.AS.2). The document marks these depth-of-knowledge levels for this indicator: Level 2 Skills of conceptual understanding.