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MCT4C1 Special Angles and the CAST RULE Many real­world applications of trigonometry deal with angle measurements of 30, 45, and 60 degrees in a right triangle. ie) Construction. Because we see these angles so much it is worth memorizing their EXACT trigonometric values! Special Triangles: To help learn these values, try memorizing the following special triangles! 60 45 2 1 1 45 30 1 Example: Using the special triangles provide EXACT answers for the following. θ 0 30 45 60 90 sinθ cosθ tanθ Related Angles: y Terminal Arm Rotational Angle θ Related/Reference Angle x Rotational Angle θ ­ The angle formed between the positive x­axis, counterclockwise to the terminal arm. Related/Reference Angle ­ the acute angle between the teminal arm and the x­axis. Example: Sketch the following angles in standard position, and state the corresponding RELATED angle. o a) 240 c) 330 o b) 135 o o d) 210 CAST rule: The CAST rule is used to determine the "sign" of a trigonometric ratio, depending on the location of the terminal arm. Each letter determines the POSITIVE ratio. Quadrant II y Quadrant I S A (sine) (all) x C T (tan) (cos) Quadrant III Quadrant IV Evaluating Trig Ratios using RELATED angles and the CAST rule: Since Trigonometric Functions are PERIODIC, the same trig ratios, repeat at regular intervals!!! Because of this we can use RELATED angles to evaluate trigonometric ratios larger than 90 degrees. Steps: Example: Find the exact value of sin210o 1.) Rewrite the trig ratio using the RELATED angle. 2.) Using the special triangles, state the EXACT value. 1.) sin210o = sin30o 2.) sin 30o = 1 2 3.) Since 210o is in Quadrant III, 3.) Use the CAST rule to determine the sign. (sine is Negative) o ∴ sin210 = ­ 1 2 Example: Find the EXACT Value of each trigonometric function. a) cos300o b) tan120o c) sin330o d) tan225o Homework: Worksheet and p. 2 ­ 3 # 4 ­ 6, 8 ­ 12