SHS2 Additional Mathematics · Semester 2 Week 5: Measurement of Triangles

Lesson notes

Learning Objectives

Indicator: 2.2.2.LI.3 - Use correct algebraic techniques to isolate the trigonometric functions and find the values for the angle that makes the equation true.

By the end of the lesson, learners can:

Solve linear trigonometric equations of the form a sin x + b = 0 (and similar forms with cos or tan) for angles in the range 0° ≤ x ≤ 360°, using inverse trigonometric functions and the CAST diagram or unit circle to find all solutions.
Factorise quadratic trigonometric equations of the form a sin²x + b sin x + c = 0 (and similar forms with cos or tan) and solve for all angles in the range 0° ≤ x ≤ 360°.
Apply algebraic techniques such as isolating the trigonometric function, factorising, and using the zero product property to reduce trigonometric equations to solvable forms.
Verify solutions to trigonometric equations by substituting the obtained angles back into the original equation, checking that each solution satisfies the equation.
Create their own linear and quadratic trigonometric equations for peers to solve, explaining their reasoning using correct mathematical vocabulary.

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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.3 - Use correct algebraic techniques to isolate the trigonometric functions and find the values for the angle that makes the equation true.
Suggested placement: Semester 2, Week 5 (Week 25 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. 371

Exemplars (from the NaCCA curriculum)

Collaborative Learning, Talk for Learning and building on what others say. Learning Experience: Learners work in mixed-ability groups to evaluate trigonometric equations.
Activity 1: Solving linear trigonometric equations Learners in mixed-ability groups solve linear trigonometric equations.
Example 1: Solve for the values of 𝑥, 2𝑠 − 1 = 0, 0° ≤ 𝑥 ≤ 360° Solution 2𝑠 − 1 = 0 1 𝑠 = 2 1
- ! 𝑥 = 𝑠 £ ¤ 2 𝑥 = 30° 𝑎 150°
Activity 2: Learners in their mixed-ability groups create linear trigonometric equations for other groups to solve and share their solutions.
Activity 3: Learners recollect how to solve quadratic equations and apply this knowledge to solve quadratic trigonometric equations.
Example 1: Solve for values of 𝑥, 2𝑠 ) 𝑥 + 𝑠 − 1 = 0, 0° ≤ 𝑥 ≤ 360°. Solution 2𝑠 ) 𝑥 + 𝑠 − 1 = 0 (2𝑠 − 1)(𝑠 + 1) = 0 2𝑠 − 1 = 0
1 𝑠 = 2 1 𝑥 = 𝑠 *! £ ¤ 2 𝑥 = 30° 𝑎 150° OR 𝑠 + 1 = 0 𝑠 = −1 𝑥 = 𝑠 *! (−1) 𝑥 = 270° ∴ 𝑥 = 30°, 150°, 270°
Teaching and Learning Resources:
- Graph paper
- Ruler
- Scientific calculator
- Protractor
- Computer
Assessment (2.2.2.AS.3). The document marks these depth-of-knowledge levels for this indicator: Level 2 Skills of conceptual understanding; Level 3 Strategic reasoning.