Maryland Math F.TF.A.3

MathGrades 9–12TRIGONOMETRIC FUNCTIONS

The Standard

Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4, and π/6 and use the unit circle to express the values of sine, cosine, and tangent for x, π + x, and 2π − x in terms of their values for x, where x is any real number.

Maryland College and Career Ready Standards for Mathematics (2025) · Extend the domain of trigonometric functions using the unit circle.

Teacher's field guide

What This Standard Means

What Students Need to Do: Students use 30-60-90 and 45-45-90 triangles to find exact sine, cosine, and tangent values. They use the unit circle to see what a half-turn or a reflection does to those values for any angle.
What Mastery Looks Like: Students find all three ratios at π/6, π/4, and π/3 from labeled triangles. They show that π+x changes (cos x, sin x) to (−cos x, −sin x), leaving tangent unchanged when defined. They show that 2π−x changes the point to (cos x, −sin x), negating tangent when defined.
Common Misconceptions: Students may swap sine and cosine for π/3 by labeling the wrong leg. Some negate both coordinates for 2π−x, rather than just the vertical coordinate. Others say tangent is zero when cosine is zero, though tangent is undefined there.
How to Assess It: Exit ticket: Draw a 30-60-90 triangle to find the unit-circle coordinates at π/6. Then find sine, cosine, and tangent at 7π/6 and 11π/6, and write the coordinate rules for π+x and 2π−x.

Lesson moves

Ways to Teach It

Cut paper special triangles with unit-length hypotenuses, place them on a unit circle, and label the legs as sine and cosine.
Have pairs explain why adding π leaves tangent unchanged when both unit-circle coordinates change sign, using a coordinate ratio.
Give pairs cards for π/6, 7π/6, and 11π/6; have them match each angle to coordinate and tangent cards.
Draw a Ferris wheel and mark a seat’s height and horizontal position at π/6, 7π/6, and 11π/6.

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