Florida B.E.S.T. MA.912.T.2.4
The Standard
Use the unit circle to express the values of sine, cosine and tangent for π-x, π+x, and 2π-x in terms of their values for x, where x is any real number.
Florida B.E.S.T. Standards for Mathematics
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students track how the point (cos x, sin x) changes after reflections and half-turns around the unit circle. They use those coordinate changes to find sine, cosine, and tangent values for related angles.
What Mastery Looks Like
- Students correctly determine the signs and values of all three functions after each angle transformation. They justify results using reflections, rotations, and the coordinates (cos x, sin x), including when tangent is undefined.
Common Misconceptions
- Students often change both signs for every transformed angle or confuse π − x with x − π. They may also forget that tangent is sine divided by cosine and is undefined when cosine is zero.
How to Assess It
- Exit ticket: If sin x = 3/5 and cos x = −4/5, find sin(π − x), cos(π + x), and tan(2π − x). Explain one answer using unit circle coordinates.
Lesson moves
Ways to Teach It
Use a paper unit circle and movable angle arm to mark x, then reflect or rotate the point for each transformed angle.
Ask students to explain why π − x keeps the sine value but changes the cosine value, using coordinates rather than memorized rules.
Give pairs of angle-expression and identity cards for students to match, then check each match on a unit circle.
Model a Ferris wheel point at x, π − x, π + x, and 2π − x, then compare horizontal and vertical coordinates.
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Related Standards
- MA.912.T.2.5
Given angles measured in radians or degrees, calculate the values of the six basic trigonometric functions using the unit circle, trigonometric identities or te...
- MA.912.T.2.2
Define the six basic trigonometric functions for all real numbers by identifying corresponding angle measures and using right triangles drawn in the unit circle...
- MA.912.T.2.3
Determine the values of the six basic trigonometric functions for 0, π/6, π/3 and π/4 and their multiples using special triangles.
- MA.912.T.2
Extend trigonometric functions to the unit circle.
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