Montana Math MT.PLUS.ALG.TRIG.1.b
The Standard
- MT.PLUS.ALG.TRIG.1
- Understand how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers by: Defining the sine and cosine functions in terms of the unit circle, and Defining the tangent, cotangent, secant, and cosecant functions in terms of sine and cosine.
Understand how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers by: Defining the tangent, cotangent, secant, and cosecant functions in terms of sine and cosine.
2026 Montana Mathematics Content Standards · Algebraic and Functional Reasoning
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students locate angles on the unit circle, including negative angles and angles beyond one turn. For a point (x, y), they write tangent as y/x, cotangent as x/y, secant as 1/x, and cosecant as 1/y. They identify when a zero denominator makes a ratio undefined.
What Mastery Looks Like
- Given 5π/6 or -π/2, a student finds the point on the unit circle and calculates each ratio that is defined. They mark undefined values and explain which coordinate is zero.
Common Misconceptions
- Students may swap secant and cosecant, or confuse cotangent with tangent. They may write 0 instead of undefined when a denominator is zero. Some treat a negative angle as a different point from its coterminal positive angle.
How to Assess It
- Exit ticket: At -π/2, the unit circle point is (0, -1). Find all four ratios, label any undefined values, and explain why they are undefined.
Lesson moves
Ways to Teach It
Place a straw from the center of a paper unit circle to (3/5, 4/5); use its endpoint to find all four ratios.
Ask: At π/2, why is cotangent 0 but tangent undefined? Write an answer using the point (0, 1).
Give students angle cards and unit-circle point cards to match, then have them write all four ratios and circle zero denominators.
Trace a bicycle wheel on paper, mark spoke positions at 45° and 90°, then find each spoke's slope as tangent.
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