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Understanding Linear Trigonometric Equations: Key Techniques and Steps

This review focuses on solving linear trigonometric equations using key triangles and the CAST rule. We explore essential first quadrant angles (0°, 30°, 45°, 60°, and 90°) and how to determine related acute angles in any quadrant. The process includes identifying signs of trigonometric ratios, drawing diagrams, and calculating principal angles. We illustrate the versatility of equations like sin(θ) = -½ and provide links to online math resources for further practice. Students are encouraged to grasp the reasoning behind their solutions rather than rote memorization.

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Understanding Linear Trigonometric Equations: Key Techniques and Steps

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  1. TF.02.5 - Linear Trigonometric Equations MCR3U - Santowski

  2. (A) Review • We have two key triangles to work with in terms of determining our related acute angles and we can place a related acute angle into any quadrant and then use the CAST “rule” to determine the sign on the trigonometric ratio • The key first quadrant angles we know how to work with are 0°, 30°, 45°, 60°, and 90°

  3. (A) Review • The two triangles and the CAST “rule” are as follows:

  4. (A) Review • We can set up a table to review the key first quadrant ratios:

  5. (B) Solving Linear Trigonometric Equations • We will outline a process by which we come up with the solution to a trigonometric equation  it is important you understand WHY we carry out these steps, rather than simply memorizing them and simply repeating them on a test of quiz

  6. (B) Solving Linear Trigonometric Equations • Work with the example of sin() = -½ • Step 1: determine the related acute angle (RAA) from your knowledge of the two triangles (in this case, simply work with the ratio of ½)   = 30° or /6 • Step 2: consider the sign on the ratio (-ve in this case) and so therefore decide in what quadrant the angle must lie  quad. III or IV in this example

  7. (B) Solving Linear Trigonometric Equations • Step 3: draw a diagram showing the related acute in the appropriate quadrants • Step 4: from the diagram determine the principle angles  240° and 300° or 4/3 and 5/3 rad.

  8. (B) Solving Linear Trigonometric Equations • One important point to realize  I can present the same original equation (sin() = -½ ) in a variety of ways: • (i) 2sin() = -1 • (ii) 2sin() + 1 = 0 • (iii)  = sin-1(-½)

  9. (C) Internet Links • Solve Trigonometric Equations from Analyze Math • Solve Trigonometric Equations - Problems from Analyze Math • Introductory Exercises from U. of Sask EMR try introductory questions first, but skip those involving proving identities • Solving Trigonometric Equations - on-line math lesson from MathTV

  10. (D) Homework • AW, text page 305, Q1-5 • MHR text, p236, Q1-5 • Nelson text, p533, Q12

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