Lesson Notes By Weeks and Term v4 - SHS 2
MEASUREMENT OF TRIANGLES
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MEASUREMENT OF TRIANGLES
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Subject: Additional Mathematics
Class: SHS 2
Term: 2nd Term
Week: 7
Grade code: 2.2.2.LI.3
Strand code: 2
Sub-strand code: 2
Content standard code: 2.2.2.CS.1
Indicator code: 2.2.2.LI.3
Theme: GEOMETRIC REASONING AND MEASUREMENT
Subtheme: MEASUREMENT OF TRIANGLES
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Welcome, students! Today, we are continuing our journey into the Measurement of Triangles. So far, we have used trigonometric ratios like sine, cosine, and tangent to find unknown sides and angles. However, in more advanced applications like surveying land in our communities, designing complex structures, or even in fields like physics and engineering, we often encounter equations where the unknown value is an angle locked inside a trigonometric function. Our goal today is to learn the algebraic techniques to "free" that angle and find all its possible values. Mastering this skill is crucial for solving complex problems involving triangles and periodic phenomena.
2.1 Recap: The Four Quadrants and the ASTC Rule
Before we solve equations, we must remember how trigonometric functions behave in the four quadrants of a circle. The range we usually work in is 0° to 360°. Quadrant 1 (0° to 90°): All ratios (Sine, Cosine, Tangent) are positive. (Mnemonic: All) Quadrant 2 (90° to 180°): Only Sine is positive. (Mnemonic: Students) Quadrant 3 (180° to 270°): Only Tangent is positive. (Mnemonic: Take) Quadrant 4 (270° to 360°): Only Cosine is positive. (Mnemonic: Chemistry)
This is the ASTC rule. To find angles in different quadrants, we first find the principal value or reference angle (α), which is the acute angle the terminal side makes with the x-axis. In Quadrant 1: θ = α In Quadrant 2: θ = 180° - α In Quadrant 3: θ = 180° + α In Quadrant 4: θ = 360° - α 2.2 Solving Linear Trigonometric Equations
A linear trigonometric equation is one where the trigonometric function has a power of 1 (e.g., `sin x`, not `sin²x`).