HSF.TF.A.2: Extending Sine and Cosine to All Real Numbers with the Unit Circle
In plain English: HSF.TF.A.2 is the Common Core functions standard that asks students to explain how the unit circle extends sine, cosine and tangent from acute angles to every real number. A real number t is read as a radian measure: travel t units counterclockwise from (1, 0), or clockwise if t is negative, and the point reached is (cos t, sin t). It is usually taught in Algebra II or Precalculus.
Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
Common Core State Standards for Mathematics · Domain: Trigonometric Functions (TF) · Cluster: Extend the domain of trigonometric functions using the unit circle Also written as HSF-TF.A.2 or F-TF.2 · Official standard
In Geometry, sine, cosine and tangent are ratios of sides in a right triangle, so they only make sense for acute angles. This lesson shows students how the unit circle removes that limit. Any real number t is read as a radian measure: starting at (1, 0), travel a distance |t| along the unit circle, counterclockwise when t is positive and clockwise when t is negative. The coordinates of the point reached are cos t and sin t, and tan t is their quotient whenever the x-coordinate is not 0.
Students first check that the new definition agrees with the old one for acute angles, then use it for angles past a quarter turn, for negative inputs and for inputs larger than 2π. The goal is an explanation students can say in their own words: why a number like 100 or -2 has a sine, and why tan t fails to exist at some inputs.
Learning Objectives
By the end of this lesson, students will be able to:
Explain why the coordinates of a point on the unit circle agree with the right-triangle definitions of cosine and sine for acute angles
Locate the point P(t) on the unit circle for any real number t, including negative numbers and numbers greater than 2π
Define cos t, sin t and tan t for any real t as coordinates of P(t), and state when tan t is undefined
Determine the signs of cos t, sin t and tan t from the quadrant where P(t) lies
Explain in words how the unit circle extends the trigonometric functions to all real numbers
Prior Knowledge Required
Students should already be comfortable with:
Radian measure as arc length on the unit circle HSF.TF.A.1
Right-triangle definitions of sine, cosine and tangent HSG.SRT.C.6
Ask students to draw a right triangle and label one acute angle θ, then write sin θ, cos θ and tan θ as ratios of sides. Then pose the prompt.
Warm-Up Prompt
"A calculator in radian mode says sin 2 ≈ 0.909 and sin(-1) ≈ -0.841. Can a right triangle have an acute angle of 2 radians, or of -1 radian? If not, what could the calculator possibly mean?"
Students should notice that 2 radians is about 115°, too big to be an acute angle, and that a negative angle cannot be an angle of a triangle at all. Collect their ideas. Some will suggest "going around a circle"; keep that idea for the next phase. A negative sine is also impossible as a ratio of two lengths, which shows that some new definition is at work.
Direct Instruction20 minutes
From triangle to circle. Draw the unit circle and a point P in Quadrant I. Drop a perpendicular to the x-axis to form a right triangle with hypotenuse 1. Then sin θ = y/1 = y and cos θ = x/1 = x. So for acute angles, cosine and sine are just the coordinates of P. Use that fact as the definition for every real number:
Read t as a radian measure: t is a length of arc on the unit circle, and its sign gives the direction.
Travel from (1, 0): go |t| units along the circle, counterclockwise if t > 0 and clockwise if t < 0. For |t| > 2π, keep going around: the path wraps as many times as needed.
Name the point: call the point reached P(t) = (x, y).
Define the functions: cos t = x, sin t = y, and tan t = y/x whenever x ≠ 0.
Stress that every real number gives exactly one point, so cosine and sine now have domain all real numbers, and their values stay between -1 and 1. Tangent is defined for every t except where P(t) is (0, 1) or (0, -1). Work through the examples with Diagram 1 and Diagram 2 on display.
Acute input: the definitions agree
For t = 0.6, a calculator gives P(0.6) ≈ (0.825, 0.565). The right triangle under P has hypotenuse 1.
Equation: cos 0.6 = adjacent/1 ≈ 0.825 and sin 0.6 = opposite/1 ≈ 0.565
Three quarters of a turn
t = 3π/2: travel 3/4 of the circumference counterclockwise.
Equation: P(3π/2) = (0, -1), so cos(3π/2) = 0, sin(3π/2) = -1, tan(3π/2) is undefined
Negative input
t = -π: travel π units, half the circle, clockwise.
t = 9π/2 = 4π + π/2: two full turns, then a quarter turn.
Equation: P(9π/2) = (0, 1), so cos(9π/2) = 0 and sin(9π/2) = 1
An input that is not a multiple of π/2
t = 4 lies between π ≈ 3.14 and 3π/2 ≈ 4.71, so P(4) is in Quadrant III.
Equation: cos 4 ≈ -0.654, sin 4 ≈ -0.757, tan 4 ≈ 1.158 (cosine and sine negative, tangent positive)
After the last example, return to the warm-up. P(2) is in Quadrant II (Diagram 1), so sin 2 is positive and cos 2 is negative. P(-1) is reached by going clockwise, into Quadrant IV, so sin(-1) is negative. The calculator values were y-coordinates all along.
Guided Practice15-20 minutes
Pairs work on printed unit circles. For each input they trace the path with a finger first, then write P(t) or its quadrant and the signs of the three functions: t = -π/2 (P = (0, -1): cosine 0, sine -1, tangent undefined), t = 3π (one and a half turns: P = (-1, 0)), t = 2.5 (Quadrant II: cosine negative, sine positive, tangent negative) and t = -1 (Quadrant IV: cos(-1) ≈ 0.540, sin(-1) ≈ -0.841). Circulate and ask each pair: "Which way did you go, and how far?" Watch for students who go clockwise for positive inputs, and for calculators left in degree mode.
Independent Practice15 minutes
Students find the exact point P(t) for t = 6π (P = (1, 0)) and t = -5π/2 (P = (0, -1)), then find the quadrant and the signs of cosine, sine and tangent for t = 5 (Quadrant IV), t = -3 (Quadrant III) and t = 10 (10 - 2π ≈ 3.72, so Quadrant III). For t = 5 and t = 10 they confirm the signs with a calculator in radian mode. They finish by writing one sentence that explains why tan t is undefined for t = -5π/2.
Closure5 minutes
Exit ticket: (1) Explain, in words, how to find sin(-0.5) using the unit circle. (Go 0.5 unit clockwise from (1, 0); sin(-0.5) is the y-coordinate of that point, about -0.479.) (2) Is cos 3 positive or negative? (Negative: 3 is between π/2 and π, so P(3) is in Quadrant II.) (3) Why does sin t have a value for every real number t, while tan t does not?
Differentiation Strategies
For Struggling Students
Give a unit circle printed with arc marks every 0.5 unit and the benchmarks π/2, π, 3π/2 and 2π labeled as decimals
Have students subtract or add 2π on a calculator until the input is between 0 and 2π before choosing a quadrant
Use a colored arrow for the direction of travel: blue for positive inputs, red for negative ones
For Advanced Students
Ask students to explain why every value of sin t lies between -1 and 1, and which inputs give exactly 1
Ask students to find all t between -2π and 2π whose point P(t) lies in Quadrant II, and write the answer as intervals
Ask why the unit circle, and not a circle of radius 3, is used for the definition: what would change in the coordinates?
Assessment Guidance
What to Look For
The standard asks students to explain, not only compute. Listen for three ideas: t is a radian measure, meaning a distance along the unit circle; the sign of t tells the direction; and cosine and sine are the coordinates of the point reached. A student who can say why sin 40 exists and is positive, without a calculator, has met the standard. Watch for students who treat inputs as degrees, go clockwise for positive t, or say that tan t is 0 (instead of undefined) when x = 0.
02
Classroom Activities
3 Activities
1
Wrap the Number Line
20 minPairs
Pairs mark a number line on adding-machine tape, using the radius of a cardboard circle as the unit, and physically wrap it around the circle. The activity makes the phrase "extend to all real numbers" literal: every point on the number line lands on a point of the circle.
Procedure
Each pair gets a cardboard circle of radius 5 cm on grid paper, with the center at the origin and 1 unit = 5 cm. Mark the tape at 0, ±1, ±2, ..., ±7 units (every 5 cm)
Tape the 0 mark at (1, 0). Wrap the positive part of the tape counterclockwise around the circle and the negative part clockwise
Mark where 1, 2, 3, 4, 5, 6 and 7 land, and where -1, -2 and -3 land. Estimate each point's coordinates from the grid
Compare the estimates with a calculator (radian mode): for example, 3 should land near (-0.99, 0.14) and 7 near (0.75, 0.66)
Discussion Questions
Why do 7 and 0.717 land at almost the same place? (7 - 2π ≈ 0.717.)
How many different numbers on the tape land on the same point of the circle?
Which direction would the tape go if the standard had said "clockwise"? Would sin 1 change sign?
Modification for Distance Learning
Use a dynamic geometry app with a slider for t that moves a point along the unit circle and shows its coordinates. Students screenshot the point for five inputs of their choice, including one negative input and one greater than 2π.
2
Stand on P(t)
15 minWhole class, in teams of 4
Tape a large unit circle and axes on the floor. Each team gets two of 8 t-value cards. A student walks from (1, 0) to P(t) in the right direction while the team calls out the quadrant and the signs of cos t, sin t and tan t.
The 8 Cards
Quadrant I: t = 1, t = 7 (7 - 2π ≈ 0.72)
Quadrant II: t = -4 (-4 + 2π ≈ 2.28), t = 15 (15 - 4π ≈ 2.43)
Quadrant III: t = 3.3, t = -2.2
Quadrant IV: t = -1.5, t = 5.8
Procedure
Before walking, the team writes a prediction: direction, number of full turns, and quadrant
The walker moves along the taped circle; a teammate checks the prediction with a calculator in radian mode
The team records cos t and sin t to two decimals and checks that the signs match the quadrant (for example, cos 5.8 ≈ 0.89 and sin 5.8 ≈ -0.46)
Challenge Variation
Teams write a new card with an input greater than 50 whose point lands in Quadrant III, and another team must verify it without walking.
3
Explain the Extension
20 minGroups of 3-4
Each group receives one "Why?" question and prepares a poster that answers it with a unit circle drawing and a worked input. Posters then go on the wall for a gallery walk, where students leave one question or correction on each.
The Four Questions
Why does the unit-circle definition give the same values as the right-triangle definition for an acute angle? Use t = 0.3.
What does a negative input mean? Use t = -2.9 and give the signs of cosine and sine.
Why are there inputs where tan t is undefined? Name two of them.
Why do cos 50 and cos(50 - 2π) have the same value? Use a calculator to show it.
Poster Requirements
A unit circle with the path from (1, 0) drawn and its direction marked
The point P(t) labeled with approximate coordinates
A three-sentence explanation that uses the words radian, arc and coordinate
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Defining Sine and Cosine for Any Real Number
The arc of length 2 runs counterclockwise from (1, 0) and ends at P(2), in Quadrant II. The dashed arc of length 1 runs clockwise and ends at P(-1), in Quadrant IV. Both points are drawn at their exact positions (cos t, sin t), to scale.
Diagram 2: Points on the Axes and Signs in Each Quadrant
Each point on an axis is reached by many inputs that differ by a whole number of full turns (multiples of 2π). The signs of cosine and sine are the signs of the x- and y-coordinates, and tangent is positive where they agree.
04
Homework Assignment
~30 min
HSF.TF.A.2 Homework: The Unit Circle and All Real Numbers
Directions: Show your work and draw a small unit circle for every part, with the path from (1, 0) and its direction marked. Set your calculator to radian mode. Give decimals to the nearest hundredth.
Part 1: Locating P(t) (Problems 1-2)
Describe the path from (1, 0) (direction and number of turns) and give the exact point P(t): (a) t = 9π (b) t = -11π/2 (c) t = 8π (d) t = -5π
Without a calculator, name the quadrant of P(t) and the signs of cos t, sin t and tan t: (a) t = 1.2 (b) t = 3.9 (c) t = -4.5 (d) t = 12
Part 2: Values from Coordinates (Problems 3-4)
Use the unit circle to find each exact value, or write "undefined": (a) cos(13π) (b) sin(-15π/2) (c) tan(10π) (d) tan(-17π/2)
The point (0.28, -0.96) is P(t) for some real number t. (a) Check that the point is on the unit circle. (b) Give cos t, sin t and tan t. (c) In which quadrant does P(t) lie? (d) What are P(t + 4π) and P(t - 2π)? Explain.
Part 3: Explaining the Extension (Problems 5-6)
Write a paragraph for a classmate who missed class, explaining how the unit circle gives a meaning to sin 100. Include what the input 100 means, about how many full turns it makes, the quadrant where P(100) lands and the sign of sin 100. Check your sign with a calculator.
A bicycle is upside down on a stand, and its wheel has radius 1 foot. Put the center of the wheel at the origin, with the valve starting at (1, 0). (a) The wheel is spun so that the valve travels 7.5 feet counterclockwise. Where is the valve now? Give its quadrant and coordinates. (b) Where would it be after traveling 7.5 feet clockwise instead?
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Locating P(t)
Direction, turns and point correct for every input
Point correct but direction or turns not shown
Most points incorrect
Signs and Values
Signs and values match the quadrant; undefined cases identified
One or two sign errors
Signs not connected to the quadrant
Explanation
Uses radian, arc and coordinate correctly to explain the extension
Explanation correct but incomplete
Missing or incorrect
Context
Wheel positions correct and interpreted
One position correct
Missing or incorrect
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Answer the questions in order and open each explanation after you choose. Keep a calculator in radian mode for the questions that ask for decimals. Reset quiz clears all answers.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
For a real number t, how do you find the point P(t) whose coordinates are cos t and sin t?
Answer: B
The standard interprets t as a radian measure traversed counterclockwise from (1, 0); a negative t means the opposite direction. Choice C reads t as degrees instead of radians. Choice D only works for acute angles and uses the wrong hypotenuse: on the unit circle it is 1.
Question 2 of 20 · Multiple Choice
Why can't the right-triangle definition sin θ = opposite/hypotenuse be used directly for θ = 2.2?
Answer: D
Right-triangle ratios exist only for acute angles, between 0 and π/2. The unit circle is what gives sin 2.2 a meaning. Choice B is false: P(2.2) is in Quadrant II, so sin 2.2 is positive (about 0.81). Choice A is irrelevant: inputs do not need to be multiples of π.
Question 3 of 20 · Multiple Choice
What is cos(-4π)?
Answer: A
Going 4π clockwise is exactly two full turns, so P(-4π) = (1, 0) and cos(-4π) = 1. Choice B would come from stopping after one and a half turns. Choice D confuses cosine with tangent, which is undefined only where x = 0.
Question 4 of 20 · Multiple Choice
What is sin(11π/2)?
Answer: C
11π/2 = 4π + 3π/2: two full turns, then three quarters of a turn, ending at (0, -1). The y-coordinate is -1. Choice A ends at (0, 1), half a turn away from the right point; it comes from removing 5π (two and a half turns) instead of a whole number of turns.
Question 5 of 20 · Multiple Choice
What is tan(-7π)?
Answer: A
Going 7π clockwise is three full turns and a half turn, ending at (-1, 0). Then tan(-7π) = y/x = 0/(-1) = 0. Choice D reverses the rule: tangent is undefined when x = 0, not when y = 0.
Question 6 of 20 · Multiple Choice
For which input is tan t undefined?
Answer: B
13π/2 = 6π + π/2, so P(13π/2) = (0, 1), where x = 0 and y/x is undefined. The other inputs are multiples of π and land on (1, 0) or (-1, 0), where tan t = 0.
Question 7 of 20 · Multiple Choice
In which quadrant is P(6)?
Answer: D
6 is between 3π/2 ≈ 4.71 and 2π ≈ 6.28, so the path from (1, 0) stops in Quadrant IV, just before completing a turn. Choice A comes from thinking 6 is more than a full turn. Check: cos 6 ≈ 0.960 and sin 6 ≈ -0.279.
Question 8 of 20 · Multiple Choice
Which statement about t = 1.9 is true?
Answer: A
1.9 is between π/2 ≈ 1.57 and π ≈ 3.14, so P(1.9) is in Quadrant II: x is negative and y is positive. Choice B is the Quadrant I pattern, a common result when students forget that 1.9 is past a quarter turn.
Question 9 of 20 · Multiple Choice
In which quadrant is P(-2)?
Answer: C
Go 2 units clockwise from (1, 0). A clockwise quarter turn is 1.57 units and a half turn is 3.14 units, so the point stops between them, in Quadrant III. Choice B comes from going counterclockwise, which gives P(2) instead.
Question 10 of 20 · Multiple Choice
Which expression is equal to cos(0.7 + 2π)?
Answer: B
Adding 2π adds one full turn to the path, so P(0.7 + 2π) is the same point as P(0.7) and the x-coordinates are equal. Choice A adds 2π to the output instead of the input; since cosine is always between -1 and 1, it cannot be right.
Question 11 of 20 · Multiple Choice
The point (-0.6, 0.8) is P(t) for some t. What is tan t?
Answer: D
tan t = y/x = 0.8/(-0.6) = -4/3. Choice A divides x by y. Choice B drops the sign: in Quadrant II the coordinates have opposite signs, so tangent is negative. Choice C is sin t.
Question 12 of 20 · Multiple Choice
Which input t gives P(t) = (0, 1)?
Answer: C
Going 7π/2 clockwise is one full turn plus three quarters of a turn clockwise, which ends at (0, 1). Equivalently, -7π/2 + 4π = π/2. Choice A lands at (0, -1), the reflection of the right point. Choice B lands at (-1, 0).
Question 13 of 20 · Multiple Choice
A calculator in degree mode gives sin 3.5 ≈ 0.061. In radian mode it gives about -0.351. Which value fits the unit-circle definition in this standard, and why?
Answer: B
The standard reads the input as a radian measure. 3.5 is between π ≈ 3.14 and 3π/2 ≈ 4.71, so P(3.5) is in Quadrant III and sin 3.5 is negative. Choice C is false for inputs past π. Choice D is false: 2π ≈ 6.28.
Question 14 of 20 · Multiple Choice
Why is sin t defined for every real number t, while tan t is not?
Answer: A
Every real number t gives exactly one point P(t), so sin t, the y-coordinate, always exists. The quotient y/x fails at (0, 1) and (0, -1), for example at t = π/2 and t = -π/2. Choice D names the wrong coordinate: y = 0 makes tan t = 0, not undefined.
Question 15 of 20 · Short Answer
Find cos t, sin t and tan t for t = -9π/2.
-9π/2 = -4π - π/2: two full turns clockwise, then another quarter turn clockwise, ending at (0, -1). So cos(-9π/2) = 0, sin(-9π/2) = -1, and tan(-9π/2) is undefined because x = 0.
Question 16 of 20 · Short Answer
Explain why the unit-circle definition of cos t and sin t gives the same values as the right-triangle definition when 0 < t < π/2.
For an acute t, P(t) = (x, y) is in Quadrant I. Dropping a perpendicular from P to the x-axis makes a right triangle with legs x and y and hypotenuse 1 (the radius). The right-triangle ratios are cos t = x/1 = x and sin t = y/1 = y, which are exactly the coordinates of P(t).
Question 17 of 20 · Short Answer
Without a calculator, give the signs of cos 5.5, sin 5.5 and tan 5.5.
5.5 is between 3π/2 ≈ 4.71 and 2π ≈ 6.28, so P(5.5) is in Quadrant IV. So cos 5.5 > 0, sin 5.5 < 0 and tan 5.5 < 0. (A calculator gives about 0.709, -0.706 and -0.996.)
Question 18 of 20 · Short Answer
Find the quadrant of P(8), then use a calculator to give cos 8 and sin 8 to the nearest thousandth.
8 - 2π ≈ 1.717, which is between π/2 ≈ 1.571 and π, so P(8) is one full turn plus a bit more than a quarter turn: Quadrant II. Calculator: cos 8 ≈ -0.146 and sin 8 ≈ 0.989, signs that match Quadrant II.
Question 19 of 20 · Short Answer
A Ferris wheel is modeled by a unit circle centered at the origin, with 1 unit equal to the wheel's radius. A car starts at (1, 0) and travels 20 units counterclockwise. Find the car's coordinates to the nearest hundredth and say how many full turns it made.
20/(2π) ≈ 3.18, so the car makes 3 full turns plus 20 - 6π ≈ 1.15 units, which is in Quadrant I. Its position is (cos 20, sin 20) ≈ (0.41, 0.91).
Question 20 of 20 · Short Answer
Explain why sin t can never be greater than 1, for any real number t.
sin t is the y-coordinate of a point on the unit circle. Every point on that circle satisfies x² + y² = 1, so y² ≤ 1 and -1 ≤ sin t ≤ 1. The largest value, 1, happens only at the top point (0, 1).
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSF.TF.A.2 mean?
It means students can explain how the unit circle lets sine, cosine and tangent take any real number as input. The number t is read as a radian measure: travel t units counterclockwise around the unit circle from (1, 0), or clockwise if t is negative. The coordinates of the point reached are cos t and sin t.
Is HSF.TF.A.2 Algebra 2 or Precalculus?
It is usually taught in Algebra II, right after radian measure (HSF.TF.A.1), and reviewed in Precalculus. It is the step that turns trigonometry from triangle ratios into functions, so it comes before graphing sine and cosine and before modeling periodic behavior (HSF.TF.B.5).
Why do we need the unit circle to define sine and cosine?
Because the right-triangle definitions only work for acute angles. A right triangle cannot have an angle of 2 radians or -1 radian. On the unit circle every real number gives a point, so sine and cosine get a value for every input, and those values agree with the triangle ratios when the angle is acute.
What does a negative angle mean on the unit circle?
A negative input means going around the circle clockwise instead of counterclockwise. For t = -2.5, start at (1, 0) and travel 2.5 units clockwise; the point is in Quadrant III, so cos(-2.5) and sin(-2.5) are both negative. The size of |t| is still the arc length.
What happens when the input is bigger than 2π?
The path wraps around the circle more than once. For t = 15, the path makes two full turns (4π ≈ 12.57) and then goes about 2.43 more units, landing in Quadrant II. Inputs that differ by a multiple of 2π land on the same point and have the same sine and cosine. The formal study of this repeating pattern, periodicity, is standard HSF.TF.A.4.
Why is tangent undefined for some numbers?
Because tan t = y/x, and x = 0 at the top and bottom of the unit circle. That happens at t = π/2 and t = -π/2 and at every input that differs from these by a multiple of π, such as 3π/2 and 5π/2. At those inputs P(t) is (0, 1) or (0, -1), and the quotient is not defined.
How do students find the sign of sin t or cos t quickly?
By finding the quadrant of P(t). Compare t, after adding or subtracting multiples of 2π, with the benchmarks π/2 ≈ 1.57, π ≈ 3.14 and 3π/2 ≈ 4.71. Then the sign of cosine is the sign of x and the sign of sine is the sign of y in that quadrant. Tangent is positive in Quadrants I and III, where x and y have the same sign.
What is a common mistake in this standard?
Leaving the calculator in degree mode. In degree mode, sin 2 means the sine of 2°, about 0.035, not the sine of 2 radians, about 0.909. Other frequent errors are going clockwise for a positive input and saying tan t = 0 where it is actually undefined.
How does this standard connect to graphing sine and cosine?
Once sin t and cos t are defined for every real t, they can be graphed as functions of t on the whole number line. The graph of y = sin t records the height of P(t) as the point moves around the circle, which is why the graph repeats every 2π and stays between -1 and 1. Graphing trigonometric functions is part of HSF.IF.C.7.
Does HSF.TF.A.2 ask students to memorize the unit circle?
No. The standard asks students to explain how the unit circle extends the functions, which is about understanding, not memorizing. Exact values for π/6, π/4 and π/3 come from special triangles in HSF.TF.A.3. For this standard, students need the points on the axes, the quadrant signs, and a clear explanation of the definition.
07
Related Standards
6 standards
These standards connect to HSF.TF.A.2: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
HSF.TF.A.1Prerequisite
Understand radian measure as arc length on the unit circle
