Big Idea:

Big Idea 3

The Unit Circle illustrates properties of trigonometric functions.

1 week

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 180o 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 30o or 45o
    • identify and justify relationships between coordinate points on the unit circle
      • Example: describe how the coordinates of 30o and  60o 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 30o or 45o
      • use reference angles to evaluate a trig expression
      • justify why the tangent of 90o and 270o 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: sin163o is between ½ and 0 because sin163o falls between sin150o and sin180o, or sin163o is between ½ and 0 because the terminal side is in Q2 where y is positive and the right triangle‘s base angle is 17o and sin17o must be less than sin30o = 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:

circle, center, radius, intercepts, domain, range, degree, special triangles, reflection, rotation, terminal side, quadrant, sine, cosine
Core Resource

No Core Resource for this Big Idea.

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