2. Using the Cartesian plane, you can find the trigonometric ratios for angles with measures greater than 90 0 or less than 0 0. Angles on the Cartesian plane are called rotational angles. An angle is in standard position when the initial arm is on the positive x-axis and the vertex is at (0, 0). Initial Arm Terminal Arm Vertex (0, 0) Angles in Standard Position

3. An angle is positive when the rotation is counterclockwise. An angle is negative when the rotation is clockwise. Quadrant I Quadrant II Quadrant III Quadrant IV Angles in Standard Position

4. Principal Angle Reference Angle is measured from the positive x-axis to the terminal arm. is measured in a counterclockwise direction, therefore is always positive. is always less than 360 0. is the acute angle between the terminal arm and the closest x-axis. is measured in a counterclockwise direction, therefore is always positive. is always less than 90 0. Angles in Standard Position

5. Principal Angle Reference Angle Principal Angle Reference Angle Principal Angle Reference Angle Angles in Standard Position

6. Sketch the following angles and list the reference and principal angles. A) 120 0 B) -120 0 C) 80 0 D) 240 0 Principal Angle Principal Angle Principal Angle Principal Angle Reference Angle Reference Angle Reference Angle Reference Angle 120 0 240 0 80 0 240 0 60 0 80 0 60 0 Finding the Reference and Principal Angles

7. Choose a point (x, y) on the terminal arm and calculate the primary trig ratios. P(x, y) x y r  r 2 = x 2 + y 2 x 2 = r 2 - y 2 y 2 = r 2 - x 2 Finding the Trig Ratios of an Angle in Standard Position

8. P(x, y) y r  x Note that x is a negative number r 2 = (x) 2 + y 2 (x) 2 = r 2 - y 2 y 2 = r 2 - (x) 2 Finding the Trig Ratios of an Angle in Standard Position Remember that in negative quadrant II, x is negative so cosine and tangent will be negative.

9. The point P(3, 4) is on the terminal arm of  List the trig ratios and find   P(3, 4) 3 4 r 2 = x 2 + y 2 = 3 2 + 4 2 = 9 + 16 = 25 r = 5 5  = 53 0 Finding the Trig Ratios of an Angle in Standard Position

10. P(-3, 4)  The point P(-3, 4) is on the terminal arm of  List the trig ratios and find  -3 4 r 2 = x 2 + y 2 = (-3) 2 + (4) 2 = 9 + 16 = 25 r = 5  ref = 53 0 5 Reference Angle 180 0 - 53 0 = 127 0 Principal Angle  = 127 0 Finding the Trig Ratios of an Angle in Standard Position

11. P(-2, 3)  -2 3 r 2 = x 2 + y 2 = (-2) 2 + (3) 2 = 4 + 9 = 13 r = √ 13  ref = 56 0 Reference Angle !! from your calculator 180 0 - 56 0 = 124 0  = 124 0 Principal Angle The point P(-2, 3) is on the terminal arm of  List the trig ratios and find  Finding the Trig Ratios of an Angle in Standard Position

12. Related angles are principal angles that have the same reference angles. These angles will also have the same trig ratios. The signs of the ratio may differ depending on the quadrant that they are in. 30 0 sin 30 0 = 0.5 PA = 30 0 PA = 150 0 PA = 210 0 Sin 150 0 = 0.5 sin 210 0 = -0.5 Related Angles

13. Using the ASTC Rule C osine A ll S ine T angent  180 0 -  Evaluate to four decimal places. A) sin 137 0 =0.6820 B) cos 142 0 = -0.7880 C) tan 158 0 = -0.4040 Find angle A, to the nearest degree: 0 0 ≤ A < 180 0 sin A = 0.341520 0 RA 20 0 160 0 cos A = -0.431864 0 RA116 0 tan A = -1.413255 0 RA 125 0 cos A = 0.632851 0 RA51 0 I II

14. C osine A ll S ine T angent  180 0 -  Find angle A, to the nearest degree: 0 0 ≤ A < 360 0 sin A = 0.5632 cos A = -0.7542 tan A = -1.5643 cos A = 0.5986 180 0 +  360 0 -  sin A = -0.8667 tan A = 0.5965 RA Quadrants III III II IV I III IV IIII 34 0 41 0 57 0 53 0 60 0 31 0 34 0 146 0 139 0 221 0 123 0 303 0 53 0 307 0 240 0 300 0 31 0 211 0 Using the ASTC Rule ( All Students Take Calculus )