THE UNIT CIRCLE Reference Angles And Trigonometry

Using Trigonometry in a Right Triangle We were limited to Acute Angles We can extend Trigonometry to Angles of Any Measure by placing those angles in the coordinate plane We do this by using reference angles, Acute Angles measured to the x-axis.

The terminal side is rotated counter-clockwise. 135° Angles are Placed with one side called the initial side on the positive x-axis.

The terminal side is rotated counter-clockwise. 135° 45° A Reference Angle is measured to the x-axis.

The terminal side is rotated counter-clockwise. 225° 45° A Reference Angle is measured to the x-axis.

The terminal side is rotated counter-clockwise. 315° 45° A Reference Angle is measured to the x-axis.

If the terminal side is rotated clockwise, the angle measure is Negative. 45° A Reference Angle is measured to the x-axis. Always Positive. -45°

Unit Circle has a radius of 1 unit.

Unit Circle has a radius of 1 unit. Cos + Sin + =y 45° x= Cosine = x Sine = y 1 45°

Cos - Sin + Cos + Sin + 135° Reference Angle = 45° 45°

135° 45° 225° 45° 45° 45° Reference Angle = 225° Cos - Sin + Cos +

135° 45° 45° 45° 45° 45° Reference Angle 315° 225° 315° Cos - Sin +

135° 45° 45° 45° 45° 45° 225° 315° Quadrant 2 Cos - Sin + Quadrant 1

Cosine = x Sine = y Tangent = Δy Δx Tangent = Sine Cosine tan = 1 45° Quadrant 1 Cos + Sin + tan = 1 Cosine = x Sine = y 45° 45° Tangent = Δy Δx Tangent = Sine Cosine

tan = -1 tan = 1 135° 45° 45° 45° 45° 45° 225° 315° tan = 1 tan = -1 Quadrant 2 Cos - Sin + Quadrant 1 tan = -1 Cos + Sin + tan = 1 135° 45° 45° 45° 45° 45° 225° 315° Quadrant 3 Cos - Sin - Cos + Sin - Quadrant 4 tan = 1 tan = -1

Tangent = Sine Cosine Quadrant 2 Cos - Sin + Tan - Quadrant 1 Cos +

Cos + Sin + 30° 30° 1 Cosine = x Sine = y

150° 150° 30° 30° 30° 30° 30° 210° 330° Cos - Sin + Cos + Sin + Cos -

Tangent = Sine Cosine 150° 30° 210° 330° Cos - Sin + Cos + Sin + Cos -

Cos + Sin + 60° 60° 1

Cos + Sin + Cos + Sin + 60° 120° 120° 60° 60°

Cos - Sin + Cos + Sin + 60° 120° 60° 60° 60° Cos - Sin - 240°

60° 120° 60° 60° 60° 60° 240° 300° Cos - Sin + Cos + Sin + Cos - Sin -

Tangent = Sine Cosine 60° 120° 240° 300° Cos - Sin + Cos + Sin + Cos -

Cosine = x Sine = y Cos(180) = -1 Cos(90) =0 Sin(180) = 0 Sin(90) = 1 90° (0 , 1) 180° Cos - Sin Cosine = x Sine = y Cos + Sin (-1 , 0) (1 , 0) Cos(0) =1 Sin(0) = 0 270° Cos(270) = 0 Sin(270) = -1 Cos Sin - (0 , -1)

Cosine = x Sine = y Tangent = Sine Cosine Cos(180) = -1 Sin(180) = 0 90° Tan(180) =0 (0 , 1) Tan(90) undefined 180° Cos - Sin Cosine = x Sine = y Cos + Sin (-1 , 0) (1 , 0) Tangent = Sine Cosine Cos(0) =1 Sin(0) = 0 270° Tan(0) =0 Cos(270) = 0 Sin(270) = -1 Cos Sin - Tan(270) undefined (0 , -1)

Evaluate the trigonometric functions at each real number. = y = x

Evaluate the six trigonometric functions at each real number. (0, -1) = y = -1 = -1 = x = 0 DNE = 0 DNE Does Not Exist

Evaluate the six trigonometric functions at each real number. -1 Sin Cos Tan Csc Sec Cot -1 So, you think you got it now?