LESSON 3 – Introducing….The CAST Rule!!! WARM-UP: Solve for r in the following triangles. Evaluate the following: a) sin25= _______ b) sin155= ________ c) tan70= _______ d) tan290= ________ b)

Terminal Arm Diagrams In order to create an angle, two rays are needed. These rays are called the initial arm and the terminal arm. Terminal arm Initial arm To construct a terminal arm diagram in standard position, the initial arm and terminal arm are placed along the positive x-axis The terminal arm is then rotated counter-clockwise according to the value of the angle, , placing it in one of the 4 quadrants.

Writing the Primary Trigonometric Ratios in Terms of x, y and r Let (x, y) be a point on the terminal arm of an angle in standard position. The side opposite to is y. The side adjacent to is x. hyp opp The hypoteneuse, r, can be determined using the Pythagorean Theorem. adj The trig ratios can be expressed in terms of x, y and r as follows:

Writing the Primary Trigonometric Ratios in Terms of x, y and r When the amount of rotation of an angle is greater than 90º, the angle is drawn as follows: represents the related angle in quadrant I. The triangle drawn in quadrant II is congruent to the triangle that would be drawn in quadrant I. and are supplementary angles.

Writing the Primary Trigonometric Ratios in Terms of x, y and r When the amount of rotation of an angle is greater than 180º or 270º the angle is drawn as follows: represents the related angle in quadrant I. The triangles drawn in all four quadrants are congruent to the triangle that would be drawn in quadrant I. How would you solve for in Quadrants III and IV??

Writing the Primary Trigonometric Ratios in Terms of x, y and r Note: All 4 triangles are congruent with the base of the triangle drawn along the x-axis. Quadrant I: Quadrant II: Quadrant III: Quadrant IV:

Based on the values and sign of x and y in each quadrant. The CAST Rule  Remember that: The CAST Rule is a quick way to see which quadrants the trig ratio is positive. A - All ratios are positive S - Only the sine ratio is positive S A T - Only the tangent ratio is positive C - Only the cosine ratio is positive T C Based on the values and sign of x and y in each quadrant.