HSF.TF.A.4: Symmetry and Periodicity of Trigonometric Functions
In plain English: HSF.TF.A.4 is an advanced (+) Common Core functions standard that asks students to use the unit circle to explain why cosine is even, sine and tangent are odd, and all three functions repeat. Reflecting the point for θ across the x-axis gives the point for -θ, and adding a full turn of 2π returns to the same point. It is usually taught in Precalculus.
(+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
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.4 or F-TF.4 · Official standard
Students use the unit circle to explain two facts about the trigonometric functions. The first is symmetry: the point for -θ is the reflection of the point for θ across the x-axis, so cos(-θ) = cos θ (cosine is even) while sin(-θ) = -sin θ and tan(-θ) = -tan θ (sine and tangent are odd). The second is periodicity: one full turn around the unit circle has length 2π, so θ and θ + 2π name the same point, and the point for θ + π is the opposite point, which is why tangent repeats every π.
The goal is explanation, not memorization. Students state each identity together with the picture that proves it, use the identities to evaluate trigonometric functions of negative and large angles, and connect the unit circle facts to the symmetry and repetition they see on the graphs.
Learning Objectives
By the end of this lesson, students will be able to:
Explain with the unit circle why the points for θ and -θ are reflections across the x-axis
Use that reflection to justify cos(-θ) = cos θ, sin(-θ) = -sin θ and tan(-θ) = -tan θ, and classify each function as even or odd
Explain why sine and cosine have period 2π and tangent has period π
Evaluate trigonometric functions of negative and large angles by using symmetry and periodicity
Connect even and odd symmetry on the unit circle to the symmetry of the graphs
Prior Knowledge Required
Students should already be comfortable with:
Radian measure as arc length on the unit circle HSF.TF.A.1
Sine and cosine of any real number as coordinates on the unit circle HSF.TF.A.2
Exact values for π/6, π/4 and π/3 and their related angles HSF.TF.A.3
Reflections of points across the x-axis and y-axis in the coordinate plane 8.G.A.3
Even and odd functions recognized from graphs HSF.BF.B.3
Hand out the unit circle sheet and give students three angles to locate:
Warm-Up Prompt
"Mark the points for π/6, -π/6 and 13π/6 on the unit circle and write the coordinates of each. Which two points are the same? How are the other two related?"
Collect answers: π/6 gives (√3/2, 1/2), -π/6 gives (√3/2, -1/2), and 13π/6 gives (√3/2, 1/2) again. Ask how students found -π/6. Many will say they went clockwise; push for the word reflection. For 13π/6, ask how many full turns they made before stopping. Tell students that these two observations, a mirror and a full turn, are the whole lesson.
Direct Instruction20 minutes
Part 1: Symmetry. Use Diagram 1. Build the argument in steps and have students copy each line next to a sketch:
The point for θ: start at (1, 0) and travel an arc of length |θ|, counterclockwise if θ > 0 and clockwise if θ < 0. The point reached is P(θ) = (cos θ, sin θ).
The point for -θ: travel the same arc length in the opposite direction. The x-axis is a line of symmetry of the circle, so P(-θ) is the reflection of P(θ) across the x-axis.
What reflection does to coordinates: reflecting across the x-axis keeps x and changes the sign of y, so P(-θ) = (cos θ, -sin θ).
Read off the identities: P(-θ) is also (cos(-θ), sin(-θ)) by definition. Comparing coordinates, cos(-θ) = cos θ and sin(-θ) = -sin θ. Then tan(-θ) = sin(-θ)/cos(-θ) = -sin θ/cos θ = -tan θ.
Name them: a function with f(-x) = f(x) is even, and one with f(-x) = -f(x) is odd. Cosine is even; sine and tangent are odd. On the graphs (Diagram 2), even means symmetric about the y-axis and odd means symmetric about the origin.
Part 2: Periodicity. The circumference of the unit circle is 2π. Traveling 2π more from P(θ) goes once around and returns to P(θ), so sin(θ + 2π) = sin θ and cos(θ + 2π) = cos θ for every θ, and the same holds for θ + 2πk for any integer k. To see that no smaller positive number p works for cosine, use θ = 0: cos(p) = cos 0 = 1 forces P(p) = (1, 0), so p is a whole number of turns. The same reasoning with θ = π/2 and the point (0, 1) works for sine. For tangent, P(θ + π) is the point diametrically opposite P(θ), namely (-cos θ, -sin θ). Both coordinates change sign, their ratio does not, and tan(θ + π) = tan θ, so tangent has period π.
Even: cosine
Find cos(-π/3) by reflecting the point for π/3 across the x-axis.
Equation: cos(-π/3) = cos(π/3) = 1/2
Odd: sine
Find sin(-3π/4). The point for -3π/4 is the reflection of the point for 3π/4.
Find tan(7π/6). The point for 7π/6 is opposite the point for π/6.
Equation: tan(7π/6) = tan(π/6) = √3/3
Symmetry and periodicity together
A calculator in radian mode gives sin(1.1) ≈ 0.891. Find sin(-1.1) and sin(1.1 + 4π) without the calculator.
Equation: sin(-1.1) ≈ -0.891 and sin(1.1 + 4π) ≈ 0.891
After the last example, point out that the identities work for every real number, not only for the special angles. That is the reason for the unit circle argument: a table of values can suggest a pattern, but the reflection and the full turn explain why it always holds.
Guided Practice15 minutes
Pairs work four problems on the unit circle sheet. For each one, they draw both points involved, name the property they used (even, odd, period 2π or period π) and give the exact value: cos(-π/4) = √2/2, sin(-5π/3) = √3/2, cos(9π/4) = √2/2 and tan(-3π/4) = 1. After each problem, one pair explains its sketch to the class. Listen for these errors: writing cos(-θ) = -cos θ because "the negative comes out", subtracting 2π once when the angle needs two full turns removed, and thinking tan(-3π/4) must be negative because the angle is negative.
Independent Practice15 minutes
Students work alone on six values and one explanation: sin(-π/2) = -1, cos(-7π/6) = -√3/2, sin(15π/4) = -√2/2, cos(20π/3) = -1/2, tan(-5π/6) = √3/3 and tan(19π/4) = -1. For each value, students write the rewritten angle they used, such as 20π/3 - 6π = 2π/3. The explanation: "Use the unit circle to explain in two sentences why sin(θ - 2π) = sin θ." Circulate and ask students who finish early to check two answers on a calculator in radian mode.
Closure5-10 minutes
Exit ticket: (1) Find sin(31π/6). (Answer: 31π/6 - 4π = 7π/6, so the value is -1/2.) (2) True or false: tan(-θ) = tan θ for every θ where both sides are defined. Explain with the unit circle. (False: the points for θ and -θ have the same x-coordinate and opposite y-coordinates, so the ratio y/x changes sign.) (3) What is the smallest positive number p with tan(θ + p) = tan θ for every θ where tan θ is defined? (π.)
Differentiation Strategies
For Struggling Students
Give a unit circle printed on tracing paper so students can fold it along the x-axis and see P(θ) land on P(-θ)
Provide a two-column organizer: left column "what happens to the point", right column "what happens to cos, sin and tan"
For large angles, have students count full turns on a number line marked in multiples of 2π before subtracting
For Advanced Students
Ask students to decide whether secant, cosecant and cotangent are even or odd and to justify each answer from the sine and cosine identities
Ask students to prove that the product of an even function and an odd function is odd, then apply it to f(x) = sin x cos x
Ask whether sin(x) + cos(x) is even, odd or neither, and to support the answer with one numerical counterexample
Assessment Guidance
What to Look For
Listen for explanations that name the reflection across the x-axis and the full turn of length 2π, not only the identity. A student who writes cos(-θ) = cos θ should be able to point to the two points on the circle and say which coordinate stayed the same. For periodicity, check that students can say why 2π is the smallest period of sine and cosine and why π is enough for tangent. In evaluation problems, look for a written rewritten angle, such as 17π/6 - 2π = 5π/6, before the final value.
02
Classroom Activities
3 Activities
1
Fold the Unit Circle
15 minPairs
Students fold a unit circle drawn on tracing paper along the x-axis and record where each point lands. The fold is the reflection behind the even and odd identities.
Procedure
Each pair marks four points on the tracing-paper unit circle: π/6, 2π/3, 5π/4, and 1 radian (about 57.3°, placed with a protractor)
Fold the sheet along the x-axis and mark where each point lands; label the new points -π/6, -2π/3, -5π/4 and -1
Record the coordinates of both points in a table. For 1 radian, use a calculator: (0.540, 0.841) and (0.540, -0.841)
Write one sentence for each column: what happened to x (cosine), what happened to y (sine), and what happened to y/x (tangent)
Discussion Questions
Why does folding along the x-axis never change the x-coordinate of a point?
Does the fold work for every angle, or only for the ones you marked? How do you know?
Which point is its own reflection? What does that say about sin 0 and sin π?
Modification for Distance Learning
Use a free online graphing calculator with a slider for θ that plots the points (cos θ, sin θ) and (cos(-θ), sin(-θ)). Students drag the slider and describe how the two points move.
2
Same or Opposite? Card Match
20 minGroups of 3-4
Groups receive 12 cards, each showing two expressions. They predict from the unit circle whether the two values are the same, opposite, or neither, then check with a calculator in radian mode. The angles are not special angles, so the prediction has to come from symmetry and periodicity.
The 12 Cards
cos(-2.3) and cos(2.3): same
sin(-0.7) and sin(0.7): opposite
tan(-1.4) and tan(1.4): opposite
sin(2 + 2π) and sin 2: same
cos(0.4 - 6π) and cos 0.4: same
tan(0.9 + π) and tan 0.9: same
sin(1.5 + π) and sin 1.5: opposite (so π is not a period of sine)
cos(-3 + 2π) and cos 3: same
sin(-5 - 2π) and sin 5: opposite
cos(2.5 + π) and cos 2.5: opposite (so π is not a period of cosine)
tan(-0.3 + 4π) and tan 0.3: opposite
sin 1 and sin 2: neither (about 0.841 and 0.909)
Procedure
Deal the cards face down. One student turns a card and predicts, with a reason that names a property (even, odd, period 2π, period π)
A second student sketches the two points on a unit circle to support or challenge the prediction
A third student checks both values on the calculator; roles rotate with each card
Groups sort the finished cards into three piles and record the reason for each card
Challenge Variation
Groups write four new cards that use secant, cosecant or cotangent, trade them with another group, and justify each answer from the sine and cosine identities.
3
Wrap the Number Line
20 minPairs
Pairs wrap a paper number line around a cardstock circle to see the real numbers land on the unit circle. Using the scale 1 unit = 5 cm, the circle has radius 5 cm and a full turn is 2π ≈ 6.28 units, about 31.4 cm of strip.
Procedure
Mark a paper strip from 0 to 13 units (65 cm), with a tick every half unit. Mark a second strip from 0 to -7 units
Tape 0 at the point (1, 0) of the circle and wrap the positive strip counterclockwise. Mark on the circle where 1, 2, 3 and 7 land
Note that 7 lands just past 1: 7 - 2π ≈ 0.72, so 7 lands at the same point as 0.72. A calculator confirms sin 7 ≈ sin(0.72) ≈ 0.657
Wrap the negative strip clockwise from (1, 0) and mark where -1 and -2 land. Compare each with the points for 1 and 2
Discussion Questions
Which numbers on your strip land on the point (1, 0)? What is the difference between neighboring ones?
Why must every number that lands on the same point as 1 have the form 1 + 2πk?
Where did -1 land compared with 1, and which identity does that show?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: The Points for θ and -θ on the Unit Circle
The point for -θ is the reflection of the point for θ across the x-axis, so it has the same x-coordinate and the opposite y-coordinate. That gives cos(-θ) = cos θ (even) and sin(-θ) = -sin θ (odd). Drawn to scale with θ = 0.9 radian.
Diagram 2: Symmetry and Period on the Graphs of Sine and Cosine
Graphs of y = sin x and y = cos x on [-2π, 2π], drawn to scale. The cosine graph is symmetric about the y-axis (even) and the sine graph is symmetric about the origin (odd). Each graph repeats every 2π because the point on the unit circle returns after one full turn.
04
Homework Assignment
~30 min
HSF.TF.A.4 Homework: Symmetry and Periodicity on the Unit Circle
Directions: Give exact values. For every value, write the property you used (even, odd, period 2π or period π) and the rewritten angle. When a problem asks you to explain, include a unit circle sketch with both points labeled.
Part 1: Even and Odd (Problems 1-2)
Use the unit circle to find cos(-2π/3), sin(-2π/3) and tan(-2π/3). Mark the points for 2π/3 and -2π/3 on your sketch.
Rewrite each expression with a positive angle, then evaluate: (a) sin(-7π/4) (b) cos(-4π/3)
Part 2: Periodicity (Problems 3-4)
Remove full turns (or half turns for tangent) to evaluate: (a) sin(29π/6) (b) cos(-17π/4) (c) tan(10π/3)
An angle b has cos b = -5/13 and sin b = 12/13. Without finding b, give cos(-b), sin(-b), sin(b + 2π), tan(-b) and tan(b - π). Name the property you used for each.
Part 3: Explaining with the Unit Circle (Problems 5-6)
A student writes sin(-θ) = sin θ because "θ and -θ have the same size." Draw the points for θ = 2π/5 and θ = -2π/5 and use them to explain the error. State the correct identity and say whether sine is even or odd.
Explain with the unit circle why tangent repeats every π while sine and cosine need 2π. Then use your explanation to evaluate tan(-23π/6).
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Symmetry
Correct even or odd identity used, with the reflection named
Correct values but no reason given
Sign errors or wrong identity
Periodicity
Correct number of full or half turns removed and shown
Correct idea with one arithmetic slip
Period not used or wrong period
Exact Values
All values exact and correct
Most values correct
Most values incorrect or decimals only
Unit Circle Explanation
Sketch with both points and a clear written reason
Sketch or reason, but not both
No explanation
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Answer each question, then open the explanation to see why the answer is right. Use exact values. Your score updates as you go, and 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
Which identity is true for every real number θ?
Answer: C
The point for -θ is the reflection of the point for θ across the x-axis, so the x-coordinate (cosine) is unchanged: cos(-θ) = cos θ. Choices A and D describe sine and tangent as even, but reflection changes the sign of y, so both are odd. Choice B describes cosine as odd, which would require the x-coordinate to change sign.
Question 2 of 20 · Multiple Choice
Find sin(-π/4).
Answer: B
Sine is odd: sin(-π/4) = -sin(π/4) = -√2/2. The point for -π/4 is (√2/2, -√2/2). Choice A treats sine as even. Choice C uses the value of sin(π/6) instead of sin(π/4). Choice D is tan(π/4).
Question 3 of 20 · Multiple Choice
Find cos(-5π/6).
Answer: D
Cosine is even: cos(-5π/6) = cos(5π/6) = -√3/2, since the point for 5π/6 is (-√3/2, 1/2). Choice A comes from treating cosine as odd and writing -cos(5π/6). Choice C uses the sine of 5π/6 with the wrong sign.
Question 4 of 20 · Multiple Choice
Find tan(-π/3).
Answer: A
Tangent is odd: tan(-π/3) = -tan(π/3) = -√3. Check with coordinates: the point for -π/3 is (1/2, -√3/2), and (-√3/2)/(1/2) = -√3. Choice B drops the sign. Choice C is the value of tan(-π/6), not tan(-π/3).
Question 5 of 20 · Multiple Choice
On the unit circle, how are the points for θ and -θ always related?
Answer: A
The same arc length traveled in opposite directions from (1, 0) gives mirror-image points across the x-axis. Choice B describes θ and π - θ. Choice C describes θ and θ + π. Choice D is true only when θ is a multiple of π.
Question 6 of 20 · Multiple Choice
What is the period of y = cos x?
Answer: C
One full turn around the unit circle has length 2π and returns to the same point, and no shorter positive turn does: cos(p) = 1 only when p is a multiple of 2π. Choice B fails because cos(x + π) = -cos x. Choice D is a period (two turns) but not the smallest one, which is what "the period" means.
Question 7 of 20 · Multiple Choice
Find sin(25π/6).
Answer: D
25π/6 - 4π = 25π/6 - 24π/6 = π/6, so two full turns are removed and sin(25π/6) = sin(π/6) = 1/2. Choice B is cos(π/6). Choice A comes from removing an odd number of half turns, 25π/6 - 3π = 7π/6, and using the sine of 7π/6; π is not a period of sine.
Question 8 of 20 · Multiple Choice
Find cos(-11π/3).
Answer: B
Cosine is even, so cos(-11π/3) = cos(11π/3). Then 11π/3 - 2π = 5π/3, and cos(5π/3) = 1/2. Adding 4π directly also works: -11π/3 + 4π = π/3. Choice A treats cosine as odd. Choice D is sin(5π/3).
Question 9 of 20 · Multiple Choice
Why does tangent have period π while sine has period 2π?
Answer: A
Adding π moves to the diametrically opposite point. Both coordinates change sign, so sin and cos change sign, but (-sin θ)/(-cos θ) = tan θ. Choice B is true but explains symmetry, not periodicity: sine is odd too and has period 2π. Choice C is false: the same point comes back only after 2π. Choice D is false: tangent is defined for all θ except odd multiples of π/2.
Question 10 of 20 · Multiple Choice
If sin t = 0.6, what is sin(-t)?
Answer: B
Sine is odd, so sin(-t) = -sin t = -0.6. Choice A treats sine as even. Choice C is the possible value of |cos t|, from 0.6² + 0.8² = 1, which answers a different question. Choice D subtracts 0.6 from 1.
Question 11 of 20 · Multiple Choice
If cos t = -0.35, what is cos(t + 6π)?
Answer: D
6π is three full turns, so t + 6π lands on the same point as t and cos(t + 6π) = cos t = -0.35. Choice B adds 6π to the output instead of the input. Choice A changes the sign, as if 6π were an odd number of half turns.
Question 12 of 20 · Multiple Choice
The graph of y = sin x is symmetric about which of the following?
Answer: C
Sine is odd, sin(-x) = -sin x, so whenever (x, y) is on the graph, (-x, -y) is too: that is symmetry about the origin. Choice A describes even functions such as cosine. Choice B is impossible for the graph of a function that is not always zero.
Question 13 of 20 · Multiple Choice
Find tan(13π/4).
Answer: D
Tangent has period π: 13π/4 - 3π = π/4, so tan(13π/4) = tan(π/4) = 1. Removing 2π instead gives 5π/4, whose point (-√2/2, -√2/2) also gives 1. Choice A comes from a sign error with the third-quadrant point. Choice B is the sine or cosine of π/4, not the tangent.
Question 14 of 20 · Multiple Choice
Which number is not a period of y = sin x?
Answer: C
A period p must satisfy sin(x + p) = sin x for every x. For p = 3π, x = π/2 gives sin(7π/2) = -1 but sin(π/2) = 1, so 3π fails: it is an odd number of half turns, and sin(x + 3π) = -sin x. Choices A, B and D are whole numbers of full turns, so each is a period; 2π is the smallest.
Question 15 of 20 · Short Answer
Use the unit circle to explain why cosine is an even function.
The point for -θ is reached by traveling the same arc length as for θ, but clockwise, so it is the reflection of the point for θ across the x-axis. A reflection across the x-axis keeps the x-coordinate. Since the x-coordinate is the cosine, cos(-θ) = cos θ for every θ, which is the definition of an even function.
Question 16 of 20 · Short Answer
Explain why sin(θ + 2π) = sin θ for every θ, and why no smaller positive number p gives sin(θ + p) = sin θ for every θ.
The unit circle has circumference 2π, so traveling 2π more from the point for θ goes once around and returns to the same point, with the same y-coordinate. For a smaller p, try θ = π/2: sin(π/2 + p) = 1 means the point for π/2 + p is (0, 1), the only point with y = 1, so p must be a whole number of full turns. The smallest positive period is 2π.
Question 17 of 20 · Short Answer
Find the exact value of sin(-19π/6). Show which properties you used.
Sine is odd: sin(-19π/6) = -sin(19π/6). Remove one full turn: 19π/6 - 2π = 7π/6, and sin(7π/6) = -1/2. So sin(-19π/6) = -(-1/2) = 1/2. Check another way: -19π/6 + 4π = 5π/6 and sin(5π/6) = 1/2.
Question 18 of 20 · Short Answer
An angle a in the first quadrant has sin a = 0.8 and cos a = 0.6. Find sin(-a), cos(-a), tan(-a) and tan(a + π).
sin(-a) = -0.8 (odd), cos(-a) = 0.6 (even), and tan a = 0.8/0.6 = 4/3. So tan(-a) = -4/3 (odd) and tan(a + π) = 4/3 (period π): the opposite point (-0.6, -0.8) gives the same ratio.
Question 19 of 20 · Short Answer
Show that tangent is odd by using the identities for sine and cosine.
For every θ where cos θ ≠ 0: tan(-θ) = sin(-θ)/cos(-θ) = (-sin θ)/(cos θ) = -tan θ. The numerator changes sign because sine is odd, and the denominator does not because cosine is even. On the unit circle, the points for θ and -θ have the same x and opposite y, so the ratio y/x changes sign.
Question 20 of 20 · Short Answer
Find every t in the interval -2π ≤ t < 4π with cos t = cos(π/5). Explain how symmetry and periodicity give all of them.
Symmetry gives two points on the unit circle with the same x-coordinate as π/5: the points for π/5 and -π/5. Periodicity adds any multiple of 2π to either. In the interval: t = -9π/5, -π/5, π/5, 9π/5, 11π/5, 19π/5. For example, 9π/5 = -π/5 + 2π and 11π/5 = π/5 + 2π.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSF.TF.A.4 mean?
HSF.TF.A.4 asks students to use the unit circle to explain why trigonometric functions are even or odd and why they repeat. Students show that the point for -θ is the reflection of the point for θ across the x-axis, which gives cos(-θ) = cos θ and sin(-θ) = -sin θ, and that a full turn of 2π returns to the same point. The "(+)" marks it as additional mathematics for students who take advanced courses such as Precalculus.
Is HSF.TF.A.4 taught in Algebra 2 or Precalculus?
It is usually taught in Precalculus. Because it is a (+) standard, many Algebra II courses introduce the unit circle and the period of sine and cosine but leave the formal even and odd arguments to Precalculus or a trigonometry course. Some integrated Math 3 courses include parts of it.
Why is cosine even and sine odd?
Because reflecting across the x-axis keeps x and flips y. The point for -θ is the reflection of the point for θ across the x-axis. Cosine is the x-coordinate, so it does not change: cos(-θ) = cos θ. Sine is the y-coordinate, so it changes sign: sin(-θ) = -sin θ. The names match power functions: x² is even and x³ is odd for the same reason.
Which of the six trig functions are even and which are odd?
Cosine and secant are even; sine, tangent, cosecant and cotangent are odd. Secant is 1/cos θ, so it inherits evenness from cosine. The other four are built with sine in the numerator or the denominator, or as a ratio of an odd and an even function, and a ratio or product of an odd and an even function is odd.
What are the periods of sine, cosine and tangent?
Sine and cosine have period 2π; tangent has period π. Any multiple of these is also a period (4π works for sine), but "the period" means the smallest positive one. In degrees, the periods are 360° and 180°.
Why is the period of tangent π instead of 2π?
Half a turn already gives the same tangent. Adding π moves to the opposite point on the unit circle, (-cos θ, -sin θ). Both coordinates change sign, and a ratio of two numbers that both change sign stays the same, so tan(θ + π) = tan θ. Sine and cosine do change sign after half a turn, so they need the full 2π.
How do you find the sine or cosine of a large or negative angle?
Use periodicity to remove full turns, and symmetry to handle the sign. For sin(41π/6), subtract 6π = 36π/6 to get 5π/6, so the value is 1/2. For cos(-3π/4), use evenness to write cos(3π/4) = -√2/2. For tangent, you may remove half turns, since its period is π. Always write the rewritten angle so the work can be checked.
What mistakes do students make with even and odd trig identities?
A common one is "pulling out the negative" from every function, writing cos(-θ) = -cos θ. Another is confusing the identity for -θ with the one for π - θ, which comes from a reflection across the y-axis instead. Students also sometimes use π as a period of sine. A quick sketch of both points on the unit circle catches all three errors.
Does HSF.TF.A.4 require a proof?
It requires an explanation, not a formal two-column proof. Students should be able to say, with a labeled unit circle sketch, why the points for θ and -θ are reflections across the x-axis and why adding 2π returns to the same point, and then connect each fact to the identities. Checking a few values on a calculator supports the explanation but does not replace it.
How does HSF.TF.A.4 connect to later topics?
Periodicity is the reason trigonometric functions can model repeating situations such as tides and rotation (HSF.TF.B.5). The even and odd identities are used to derive the subtraction formulas from the addition formulas (HSF.TF.C.9), to simplify expressions, and in calculus, where symmetry about the origin makes the integral of an odd function over [-a, a] equal to zero.
07
Related Standards
5 standards
These standards connect to HSF.TF.A.4: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
HSF.TF.A.2Prerequisite
Use the unit circle to extend trigonometric functions to all real numbers
