LESSON 5-3: Evaluating Trigonometric Ratios of Angles Greater than 90o Placing Angles on the Cartesian X-Y Plane y << << x << << A PRINCIPAL angle, is formed by rotating a ray about a fixed point called the The ray is called the at the beginning of the angle and the at the end of the angle. An angle, is in if : the vertex of the angle is at the origin, the initial arm lies fixed along the positive x-axis, and the terminal arm is the final position of the rotating ray anywhere on the arc of rotation. Trig Ratios of Principle Angles Rotating a point on the circumference of a circle creates a principle angle, . The side opposite is and the side adjacent to is . We choose as the hypotenuse since it represents the radius of a circle. y P(x, y) x TRIGONOMETRIC ratios can be defined in terms of the x and y co-ordinates of any point P(x, y): If r 2 = x2 + y2 , r = and r > 0, then: 1. sin = csc = 2. cos = sec = 3. tan = cot = . Angle Direction : An angle can have a positive or negative value: y y x A positive angle is formed by a counter-clockwise rotation of the terminal arm. A negative angle is formed by a clockwise rotation of the terminal arm. Coterminal Angles : Angles in standard position that have the same TERMINAL arm. Ex.1: State two coterminal angles for a) 30 and for b) 120 . x Investigation 1: The “CAST” Rule Complete the table below to find the SIGN of each trig ratio for the angle in each quadrant: I in QUADRANT: II III IV y r x cos r y tan x II I sin III Which ratio is +ve in this quadrant? IV Conclusion: The “CAST” Rule determines which trig ratio is POSITIVE in each quadrant. Investigation 2: Trig Ratios of Angles Greater than 90˚ 1.Complete the table below: 30˚ Angle, sin cos tan 150˚ 210˚ 330˚ 390˚ -30˚ 2. Draw the angles above on the Cartesian Plane: Related acute angle( ): Ex. If = 240°, then = Conclusions: 1. If the principal angle, . . , is greater than 90 o, then a , is formed. 2. The related acute angle, , and the angles greater than 90 o can be used to find trig ratios of Ex. 1: The point (–6, 8) is on the terminal arm of an angle in standard position. a) Determine the primary trig ratios for the principal angle, . b) What is the measure of ? Ex. 2: Find all possible measures of angle to the nearest degree if 0 360. Use CAST. a) sin = 0.4226 b) cos = - 4 5 c) cot = Ex. 3: Find the exact value of the sine, cosine, and tangent of if: a) = 240o b) = 315o Ex. 4: Use the point P(0,1) to determine the primary trig ratios for a 90 angle. Homework: p. 281 #2, 3odds, 5, 6, 10, 11 p. 348 #1-3 (tan ratio only) 5, 6c, 7ce, 11fij,12,13b, 17 3 Reviewing Trig Ratios of Angles Greater than 90˚ Recall: Given a point on the terminal arm of any angle in the x-y plane, P(x,y), we may find the primary trig ratios for the principal angle, and the measure of that angle as follows: 1. Create a right triangle between the terminal arm and the closest x-axis. P(x,y) 2. Label x, y, r and the related acute angle,β, in this right angle triangle. 3. Find the required trig ratio for the related acute angle, β using: y x y sin cos tan . r r x and then determine the measure of angle . 4. Find the measure of the principal angle, by: (i) using the CAST Rule to determine # of possible answers (ii) add or subtract to 180 or to 360 to determine the principal angle . Quadrant I II III IV Principal Angle Measure, 180 - 180 + 360 -