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

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

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 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 Angles in Standard Position

Principal Angle Reference Angle Principal Angle Reference Angle Principal Angle Reference Angle Angles in Standard Position

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

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

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.

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 = = = 25 r = 5 5  = 53 0 Finding the Trig Ratios of an Angle in Standard Position

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 = = 25 r = 5  ref = Reference Angle = Principal Angle  = Finding the Trig Ratios of an Angle in Standard Position

P(-2, 3)  -2 3 r 2 = x 2 + y 2 = (-2) 2 + (3) 2 = = 13 r = √ 13  ref = 56 0 Reference Angle !! from your calculator =  = 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

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 sin 30 0 = 0.5 PA = 30 0 PA = PA = Sin = 0.5 sin = -0.5 Related Angles

Using the ASTC Rule C osine A ll S ine T angent   Evaluate to four decimal places. A) sin = B) cos = C) tan = Find angle A, to the nearest degree: 0 0 ≤ A < sin A = RA cos A = RA116 0 tan A = RA cos A = RA51 0 I II

C osine A ll S ine T angent   Find angle A, to the nearest degree: 0 0 ≤ A < sin A = cos A = tan A = cos A =   sin A = tan A = RA Quadrants III III II IV I III IV IIII Using the ASTC Rule ( All Students Take Calculus )