Introduction to Unit Circle Trigonometry 1. The Unit Circle on the Coordinate Plane (1,0) (0,1) (-1,0) (0, -1) Quadrant I X – Pos Y - Pos X Y Radius =

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Presentation transcript:

Introduction to Unit Circle Trigonometry 1. The Unit Circle on the Coordinate Plane (1,0) (0,1) (-1,0) (0, -1) Quadrant I X – Pos Y - Pos X Y Radius = 1 Quadrant II X – Neg Y - Pos Quadrant IV X – Pos Y - Neg Quadrant III X – Neg Y - Neg

Introduction to Unit Circle Trigonometry 1. The Unit Circle on the Coordinate Plane 0° Circumference = 360° Y 90° 180° 270° 360° Position at 0° and 360° is the same. Full rotation or come full circle. Ferris wheel, back where you started

Introduction to Unit Circle Trigonometry 2. Converting Between Degrees and Radians 360° = 2π radians ÷360° 1 = _π. 180° Ex 1: Convert from degrees to radians a.90° b.45° c.135° d.180° e.-60° Simplify Fractions NO DECIMALS ● _π. = 180° _π_ 2 ● _π. = 180° π_ 4 ● _π. = 180° _3π_ 4 ● _π. = 180° π ● _π. = 180° 3 1 _-π_ 3 D R

Introduction to Unit Circle Trigonometry 3. Converting Between Radians and Degrees 360° = 2π radians ÷2π 180°_ = 1 π Ex 2: Convert from radians to degrees a.2π b.π 2 c. 3π 2 d. 5π 6 Simplify Fractions NO DECIMALS ● 180° = π 1 360° ● 180° = π 90° 1 90 ● 180° = π 270° 1 90 ● 180° = π 150° 1 30 R D

Introduction to Unit Circle Trigonometry 1. The Unit Circle on the Coordinate Plane 0° 2π = 360° Y 90° 180° 270° 360° 0 radians π2π2 2π = π 2 3π23π2 4π = 2π 2

Introduction to Unit Circle Trigonometry 4. Angle Measures 0° Y 90° 180° 270° 360° ? 30° = Angle 120° Ex 3 Find the measure of the indicated angle

Introduction to Unit Circle Trigonometry 0° Y 90° 180° 270° 45° = 225° Ex 4 Find the measure of the indicated angle ? 4. Angle Measures

Introduction to Unit Circle Trigonometry 0° Y 90° 180° 270° 45° = 225° 225° - 360° = -135° Ex 5 Find the measure of the indicated angle ? 360° 4. Angle Measures

Introduction to Unit Circle Trigonometry Y ? π6π6 π + π = 6 2 π + 3π = 4π = 2π Ex 6 Find the measure of the indicated angle 0 radians π2π2 2π = π 2 3π23π2 4π = 2π 2 Simplify Fractions NO DECIMALS 2π ● 180° = 120° 3 π Check w Ex 4 4. Angle Measures

Introduction to Unit Circle Trigonometry Y π + π = 1 4 4π + π = 5π π - 2π = 4 5π - 8π = -3π Ex 7 Find the measure of the indicated angle π2π2 2π = π 2 3π23π2 4π = 2π 2 Simplify Fractions NO DECIMALS -3π ● 180° = -135° 4 π Check w Ex 5 ? π4π4 4. Angle Measures

Introduction to Unit Circle Trigonometry 2 full rotations = 360*2 = 720° + partial rotation Partial rotation = 180° - 75° = 105° Total = = 825° Ex 8 Find the measure of the indicated angle 75° Every rotation counts as 360° or 2π radians 4. Angle Measures

Introduction to Unit Circle Trigonometry 2 full rotations = 2π*2 = 4π + partial rotation Partial rotation = π - π = 3 3π - π = 2π Total = 4π + 2π = 12π + 2π = 14π radians Ex 9 Find the measure of the indicated angle π3π3 Every rotation counts as 360° or 2π radians 4. Angle Measures

Introduction to Unit Circle Trigonometry 2 full negative rotations = -360*2 = -720° + partial rotation Partial rotation = -30° Total = = -750° Ex 10 Find the measure of the indicated angle -30° Every rotation counts as 360° or 2π radians 4. Angle Measures

Introduction to Unit Circle Trigonometry Ex 11 Find the measure of the indicated angle Every rotation counts as 360° or 2π radians 2 full negative rotations = -2π*2 = -4π + partial rotation Partial rotation = - π 3 Total = -4π + -π = 3 -12π + -π = -13π radians Angle Measures