Evidence of Understanding

• explain patterns in the unit circle
  • graph the unit circle and identify its center, radius, intercepts, domain, and range
  • use the unit circle’s radius to explain why 180^o is equivalent to π
  • create diagrams that illustrate equivalent degree and radian measures
    • describe how degrees and radians are related
    • convert between degrees and radians
    • approximate the degree measure for a single radian  
  • use transformations and special triangles to find points on the unit circle for central angles that are multiples of 30^o or 45^o
  • identify and justify relationships between coordinate points on the unit circle
    • Example: describe how the coordinates of 30^o and  60^o are related, etc.
  • describe the relationship between the angle measure and the sign value of its coordinates
    • Example: any angle with a terminal side in the second quadrant has coordinates (-x, + y)
  • prove the Pythagorean Identity sin2θ + cos2θ = 1
    • justify coordinates on the unit circle as (cosθ, sinθ)
      
• use the unit circle to extend understanding of trigonometry
  • justify intervals when values of sinθ or cosθ increase or decrease from a diagram or table
  • explain why the sine or cosine of any angle must be between 1 and -1, inclusively
  • determine the exact value of any sine, cosine, tangent, cosecant, secant, or cotangent expression for angles that are multiples of 30^o or 45^o
    • use reference angles to evaluate a trig expression
    • justify why the tangent of 90^o and 270^o are undefined
    • explain why the tangent values are positive in quadrants 1 and 3
  • approximate the value of any sin, cos, tan, csc, sec, or cot expression from the unit circle
    • Example: sin163^o is between ½ and 0 because sin163^o falls between sin150^o and sin180^o, or sin163^o is between ½ and 0 because the terminal side is in Q2 where y is positive and the right triangle‘s base angle is 17^o and sin17^o must be less than sin30^o = ½
  • use properties of the unit circle to solve for an unknown angle in a trig equation

Develop conceptual understanding:

unit circle, radian, Pythagorean Identity, tangent, cosecant, secant, cotangent, undefined

Supporting terms to communicate: