SECTION 2.1 EQ: How do the x- and y-coordinates of a point in the Cartesian plane relate to the legs of a right triangle?

Warm-Up/Activator Name the quadrants of the Coordinate (Cartesian) Plane – use drawing on next slide. Label the signs of x and y in each quadrant If angle measurement travels in the same direction as the naming of the quadrants, which direction (clockwise or counter- clockwise) is the positive direction? Given that the positive end of the x-axis is the initial side of an angle, and therefore 0˚, label the corresponding angles on the other three ends of the x- and y-axis.

Drawing for the Warm-up/Activator

Coterminal Angles Two angles in standard position are coterminal if they have the same __________ ______.

Example 2 Determine whether the following pairs of angles are coterminal. a) b)

Your Turn 2 Determine whether the following pairs of angles are coterminal. a) b)

Example 3 Determine the angle of the smallest possible positive measure that is coterminal with each of the following angles. a) 830˚b) -520˚

Your Turn 3 Determine the angle of the smallest possible positive measure that is coterminal with each of the following angles. a) 900˚b) -430˚

REFERENCE ANGLES How do angles in quadrant II, III and IV relate to angles in quadrant I?

Vocabulary Terminal Side: the rotating ray of an angle Quadrantal Angles: an angle whose terminal side lies along the x- or y-axis. Reference angle: acute angle formed by the terminal side and the x-axis

Terminal side 180° 0° 90° 270° 360° + - I II III IV

Angle Θ = 115° To find the reference angle when the angle is in Quadrant II, subtract the angle from 180°. Reference Angle 180° – 115° = 65°

Θ = 225° To find the reference angle when the angle is in Quadrant III, subtract 180° from the angle. Reference Angle 225° – 180° = 45°

Θ = 330° To find the reference angle when the angle is in Quadrant IV, subtract the angle from 360°. Reference Angle 360° – 330° = 30°

Θ = -150° Reference Angle 360° + – 150° = 210° 210° – 180° = 30°

Example 1 Sketch the following angles in standard position. State the quadrant in which (or axis on which) the terminal side lies. Also state the reference angle. a) -90˚b) 210˚

Your Turn 1 Sketch the following angles in standard position. State the quadrant in which (or axis on which) the terminal side lies. Also state the reference angle. a) -300˚b) 135˚

ANALYZING VALUES OF THE UNIT CIRCLE Why do we analyze the values of the special angles in all quadrants of the unit circle?

Special Triangles 60° ° Sin 30 Cos 30 Tan 30 Sin 60 Cos 60 Tan 60

45° 1 1 Special Triangles Sin 45 Cos 45 Tan 45

ϑ 1 x y P(cosϑ, sinϑ) P (x,y) The unit circle is a circle with a radius of one

Θ 1 P(cosΘ, sinΘ) 1 0 ° (1,0) 180 ° (-1,0) 90 ° (0,1) 270 ° (0,-1) Values of the quadranal angles: What are quadranal angles? The angles whose terminal sides are on the axes. Continue on the top circle:

30 0°0° 30 ° 150 ° 330 ° 210 ° 360 ° 180 ° 270 ° 90 °

45 0°0° 45 ° 135 ° 315 ° 225 ° 360 ° 180 ° 270 ° 90 °

60 0°0° 60 ° 120 ° 300 ° 240 ° 360 ° 180 ° 270 ° 90 °

°0° 30 ° 150 ° 330 ° 210 ° 45 ° 135 ° 315 ° 225 ° 60 ° 120 ° 300 ° 240 ° 360 ° 180 ° 270 ° 90 °

Unit Circle

Using the Unit Circle to find exact sin and cos values Go back to Examples 2 & 3 and determine the exact sin and cos values for the angles.