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HSF.TF.C.9Common CoreMathFunctionsGrades 9-12

HSF.TF.C.9: Proving and Using the Addition and Subtraction Formulas

In plain English: HSF.TF.C.9 is an advanced (+) Common Core functions standard, usually taught in Precalculus. Students prove the angle addition and subtraction formulas for sine, cosine and tangent, starting from a distance argument on the unit circle, and use them to find exact values such as cos 105°, to simplify expressions and to solve equations and applied angle problems.

(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Common Core State Standards for Mathematics · Domain: Trigonometric Functions (TF) · Cluster: Prove and apply trigonometric identities
Also written as HSF-TF.C.9 or F-TF.9 · Official standard
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01

Lesson Plan

70-80 min

Overview

Students prove the addition and subtraction formulas for sine, cosine and tangent. The lesson starts with the cosine of a difference, proved by computing the same chord length on the unit circle in two ways, and then derives the other five formulas from it using even and odd properties, cofunctions and the definition of tangent.

Students then use the formulas to find exact values of angles such as 105° and π/12, to combine given values of two angles in different quadrants, to simplify expressions, and to solve equations and angle problems. Each problem starts with choosing the formula and its sign pattern.

Learning Objectives

By the end of this lesson, students will be able to:

  • Prove the formula for cos(α - β) with a distance argument on the unit circle
  • Derive the formulas for cos(α + β), sin(α ± β) and tan(α ± β) from the cosine difference formula
  • Use the formulas to find exact values and to combine given trigonometric values of two angles
  • Use the formulas to simplify expressions, prove short identities, and solve equations and applied angle problems

Prior Knowledge Required

Students should already be comfortable with:

  • The Pythagorean identity sin²θ + cos²θ = 1 and finding values from the quadrant HSF.TF.C.8
  • Exact values for π/6, π/4 and π/3 from special triangles HSF.TF.A.3
  • Even and odd symmetry of sine and cosine on the unit circle HSF.TF.A.4
  • The distance formula in the coordinate plane

Lesson Procedure

70-80 minutes of class time across 5 phases.

  1. Warm-Up10 minutes

    Write one question on the board and have students answer with exact values they already know:

    Warm-Up Prompt

    "Is sin(30° + 60°) equal to sin 30° + sin 60°? Is cos(90° - 30°) equal to cos 90° - cos 30°? What does this tell you about trigonometric functions?"

    sin 90° = 1, but sin 30° + sin 60° = (1 + √3)/2 ≈ 1.37. cos 60° = 1/2, but cos 90° - cos 30° = -√3/2. Sine and cosine do not distribute over addition. Tell students that there are correct formulas for sin(α + β) and cos(α - β), that each one uses both sine and cosine, and that they will prove them before using them.

  2. Direct Instruction25 minutes

    Part 1: The proofs. Use Diagram 1 for the key step and build the other five formulas from it:

    1. Cosine of a difference: place A = (cos α, sin α) and B = (cos β, sin β) on the unit circle. By the distance formula and the Pythagorean identity, AB² = (cos α - cos β)² + (sin α - sin β)² = 2 - 2(cos α cos β + sin α sin β).
    2. Rotate: turn the circle by -β. B moves to D = (1, 0) and A moves to C = (cos(α - β), sin(α - β)). Rotation keeps distances, so CD = AB, and CD² = (cos(α - β) - 1)² + sin²(α - β) = 2 - 2cos(α - β). Setting the two equal gives cos(α - β) = cos α cos β + sin α sin β.
    3. Cosine of a sum: replace β with -β. Because cos(-β) = cos β and sin(-β) = -sin β, cos(α + β) = cos α cos β - sin α sin β.
    4. Sine of a sum and difference: sin(α + β) = cos(π/2 - (α + β)) = cos((π/2 - α) - β). Apply the difference formula and the cofunction facts cos(π/2 - α) = sin α and sin(π/2 - α) = cos α: sin(α + β) = sin α cos β + cos α sin β. Replacing β with -β gives sin(α - β) = sin α cos β - cos α sin β.
    5. Tangent: tan(α + β) = sin(α + β)/cos(α + β). Divide the numerator and denominator by cos α cos β: tan(α ± β) = (tan α ± tan β)/(1 ∓ tan α tan β), defined when the tangents and the denominator are defined.

    Diagram 2 gives a second, geometric proof of the sum formulas for acute angles, which many students find easier to remember. Part 2: Using the formulas. Work through these examples. Before each one, ask which formula fits and how to split the angle into known angles.

    • Cosine of a sum, exact value

      Find cos 105° using 105° = 60° + 45°.

      Equation: cos 60° cos 45° - sin 60° sin 45° = (√2 - √6)/4

    • Sine of a difference, exact value

      Find sin(π/12) using π/12 = π/3 - π/4.

      Equation: sin(π/3)cos(π/4) - cos(π/3)sin(π/4) = (√6 - √2)/4

    • Tangent of a sum, exact value

      Find tan 75° using 75° = 45° + 30°.

      Equation: (1 + √3/3)/(1 - √3/3) = 2 + √3

    • Given values and quadrants

      sin α = 3/5 with α in Quadrant I, cos β = -5/13 with β in Quadrant II. Find sin(α + β) and cos(α + β).

      Equation: cos α = 4/5, sin β = 12/13; sin(α + β) = 33/65 and cos(α + β) = -56/65

    • Solving an equation

      Solve sin x cos(π/3) + cos x sin(π/3) = 1 for 0 ≤ x < 2π.

      Equation: sin(x + π/3) = 1, so x + π/3 = π/2 and x = π/6

  3. Guided Practice15 minutes

    Pairs work four problems and compare after each. (1) sin 75° = sin(45° + 30°) = (√6 + √2)/4. (2) cos(π/12) = cos(π/3 - π/4) = (√2 + √6)/4. (3) tan 15° = tan(45° - 30°) = 2 - √3. (4) tan α = 2 and tan β = 3, both in Quadrant I: tan(α + β) = 5/(1 - 6) = -1, so α + β = 135°. For each problem, pairs first write the formula with the correct sign pattern. Listen for these errors: the plus sign in cos(α + β), splitting 75° into angles without known values, and a wrong sign in the tangent denominator.

  4. Independent Practice15 minutes

    Students work alone: (1) sin 195° = sin(150° + 45°) = (√2 - √6)/4. (2) cos 165° = cos(120° + 45°) = -(√2 + √6)/4. (3) tan 105° = tan(60° + 45°) = -2 - √3. (4) Prove that cos(π/2 - x) = sin x using the difference formula. (5) cos α = 12/13 with α in Quadrant I and sin β = -4/5 with β in Quadrant IV: find sin(α - β) (63/65). Students check one exact value in problems 1-3 with a calculator.

  5. Closure5-15 minutes

    Exit ticket: (1) Use a subtraction formula to show that cos(x - π) = -cos x. (2) Write tan(π/4 + x) in terms of tan x. (Answer: (1 + tan x)/(1 - tan x).) Use the longer time if students present their proofs from the independent practice.

Differentiation Strategies

For Struggling Students

  • Give a formula card with the six formulas color-coded: sine keeps the sign and mixes the functions, cosine changes the sign and keeps the functions together
  • Provide a table of angles that are sums or differences of 30°, 45° and 60°, so students choose the split before using a formula
  • Walk through the distance proof with α = 110° and β = 40° and a calculator before the general version

For Advanced Students

  • Ask students to derive sin 2θ = 2 sin θ cos θ and cos 2θ = cos²θ - sin²θ from the sum formulas
  • Ask students to write 3 sin t + 4 cos t in the form R sin(t + φ) and explain what R and φ mean for a combined wave
  • Ask students to prove the tangent formula directly from Diagram 2 and state when it fails

Assessment Guidance

What to Look For

In the proof of cos(α - β), look for the two expressions for the same distance and a reason why rotation keeps lengths. In the derivations, check that students name each fact they use: the even and odd properties, the cofunction identity, or the definition of tangent. In computations, check the sign pattern first, then the choice of angles, then the exact arithmetic, and ask for a calculator check of one decimal value.

02

Classroom Activities

3 Activities

1

Chord Proof with Measurement

20 minPairs

Pairs build the distance proof of cos(α - β) on a large unit circle, first with measured numbers and then with letters.

Procedure

  • Draw a circle of radius 10 cm (so 10 cm stands for 1 unit). Mark A at 110° and B at 40°, then C at 70° and D at 0°
  • Measure chords AB and CD with a ruler. Both should be about 11.5 cm, because 2 sin(35°) ≈ 1.147
  • Compute AB² from the coordinates of A and B with a calculator and CD² from the coordinates of C and D. Both equal 2 - 2cos 70° ≈ 1.316
  • Repeat the computation with letters, following Diagram 1, and finish with cos(α - β) = cos α cos β + sin α sin β

Discussion Questions

  • Why do AB and CD have the same length? Which transformation takes one to the other?
  • Where does the Pythagorean identity make the algebra shorter?
  • Would the proof still work if α were less than β?

Modification for Distance Learning

Use a shared dynamic geometry file with sliders for α and β. Pairs drag the sliders and record the two chord lengths for three different pairs of angles before writing the algebra.

2

Formula Family Jigsaw

20-25 minGroups of 3

Each group starts with the proved formula for cos(α - β). Each member derives a different case and teaches it to the others, so the group ends with a complete set of six proved formulas.

Roles

  • Cosine expert: derive cos(α + β) by replacing β with -β, and explain why cos(-β) = cos β and sin(-β) = -sin β
  • Sine expert: derive sin(α + β) from sin θ = cos(π/2 - θ), then sin(α - β)
  • Tangent expert: derive tan(α + β) and tan(α - β) by dividing by cos α cos β, using the two sum formulas the other experts produce

Procedure

  • Experts from different groups meet for 8 minutes to compare their derivations
  • Experts return and teach the group; every member copies all six formulas with the reason for each step
  • The group checks each formula with α = 60° and β = 30°, whose values are all known

Discussion Questions

  • Why does the tangent expert have to wait for the others?
  • For which angles is the tangent formula undefined, even though sin(α + β) and cos(α + β) exist?
3

Problem-Solving Stations

25 minGroups of 3-4

Groups rotate through four stations, about 6 minutes each. Each station uses a different formula to solve a different kind of problem.

Stations

  • Station 1, exact value: find sin 255° using 210° + 45°. (-(√2 + √6)/4)
  • Station 2, combining values: sin α = 8/17 in Quadrant II and cos β = 3/5 in Quadrant I. Find cos(α - β). (cos α = -15/17 and sin β = 4/5, so cos(α - β) = -13/85)
  • Station 3, equation: solve cos x cos(π/4) - sin x sin(π/4) = √2/2 for 0 ≤ x < 2π. (cos(x + π/4) = √2/2, so x = 0 or x = 3π/2)
  • Station 4, angle between two lines: the lines y = 3x and y = x/2 make angles α and β with the x-axis, where tan α = 3 and tan β = 1/2. Find the angle between the lines. (tan(α - β) = (5/2)/(5/2) = 1, so 45°)

Procedure

  • At each station, the group writes the formula it used before starting the computation
  • One member checks the answer with a calculator before the group moves on
  • At the end, each group explains one station to the class

Challenge Variation

At Station 4, groups find the angle between y = 2x and y = -x, and explain what a negative value of tan(α - β) says about the angle.

03

Diagrams & Visual Aids

2 diagrams

Diagram 1: The Chord Proof of cos(α - β)

A(cos α, sin α) B(cos β, sin β) C(cos(α - β), sin(α - β)) D(1, 0) O Example: α = 110°, β = 40° AB² = 2 - 2(cos α cos β + sin α sin β) CD² = 2 - 2 cos(α - β) Rotating by -β moves B to D and A to C, so AB = CD. Here AB = CD = 2 sin 35° ≈ 1.147 cos(α - β) = cos α cos β + sin α sin β
Drawn to scale with α = 110° and β = 40°. Chord AB joins the points for α and β; chord CD joins the points for α - β = 70° and 0°. A rotation by -β takes B to D and A to C, so the chords have the same length. Writing both lengths with the distance formula and setting them equal proves cos(α - β) = cos α cos β + sin α sin β.

Diagram 2: A Rectangle Proof of the Sum Formulas

O Q P cos β sin β 1 α β cos α cos β sin α cos β cos α sin β sin α sin β cos(α + β) sin(α + β) Drawn for α = 35°, β = 25° Left side = right side: sin(α + β) = sin α cos β + cos α sin β Top side = bottom side: cos(α + β) = cos α cos β - sin α sin β
Drawn to scale with α = 35° and β = 25°. The right triangle OQP has hypotenuse OP = 1 and an angle β at O, so OQ = cos β and QP = sin β. The smaller right triangles give the side pieces shown. Opposite sides of the rectangle are equal, which gives both sum formulas for acute angles with α + β < 90°.

04

Homework Assignment

~30 min

HSF.TF.C.9 Homework: Addition and Subtraction Formulas

Directions: Show every step and name the formula you use. Give exact answers in simplest radical form. In each proof, give a reason for every step. Check at least one exact value with a calculator.

Part 1: Proofs (Problems 1-2)

  1. Starting from sin(α + β) = sin α cos β + cos α sin β, prove sin(α - β) = sin α cos β - cos α sin β. Name the property of sine and cosine you use. Then check the formula numerically with α = π/3 and β = π/4.
  2. Starting from the formulas for sin(α - β) and cos(α - β), prove that tan(α - β) = (tan α - tan β)/(1 + tan α tan β). State the conditions on α and β under which your proof is valid.

Part 2: Exact Values and Given Ratios (Problems 3-4)

  1. Find the exact value of each expression: (a) cos 255° (b) sin(11π/12) (c) tan 165°.
  2. sin α = 7/25 with α in Quadrant II, and cos β = 8/17 with β in Quadrant IV. Find the exact values of cos(α + β), sin(α - β) and tan(α + β).

Part 3: Solving Problems (Problems 5-6)

  1. Use the addition and subtraction formulas to simplify sin(x + π/6) - sin(x - π/6). Then solve sin(x + π/6) - sin(x - π/6) = 1/2 for 0 ≤ x < 2π.
  2. A mural on a wall starts 3 m above a viewer's eye level and ends 8 m above it. The viewer stands 20 m from the wall. The angle to the top of the mural is α, with tan α = 8/20, and the angle to the bottom is β, with tan β = 3/20. Use a tangent formula to find tan(α - β), the tangent of the viewing angle, as an exact fraction, and then find the viewing angle to the nearest tenth of a degree.

Rubric

CriterionFull Credit (2 pts)Partial Credit (1 pt)No Credit (0 pts)
ProofsEvery step follows from a named property or formulaCorrect result with a missing reasonResult stated without proof
Formula Choice and SignsCorrect formula and sign pattern in every problemOne sign errorWrong formula or sine and cosine distributed
Exact ValuesAll values exact, simplified and checked with a calculatorCorrect but not simplified or not checkedValues missing or decimal only
Solving ProblemsAll solutions in the interval found and the context answeredOne solution missing or angle not interpretedNo valid solution

05

Quiz: 20 Questions

Interactive, with answers

Instructions

Give exact answers where you can. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.

Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.

0 of 20 answered · 0 correct

  1. Question 1 of 20 · Multiple Choice

    Which is the formula for sin(α + β)?

  2. Question 2 of 20 · Multiple Choice

    Which is the formula for cos(α - β)?

  3. Question 3 of 20 · Multiple Choice

    Which expression equals tan(x - π/4)?

  4. Question 4 of 20 · Multiple Choice

    What is the exact value of cos 75°?

  5. Question 5 of 20 · Multiple Choice

    What is the exact value of sin 285°?

  6. Question 6 of 20 · Multiple Choice

    tan α = 1/2 and tan β = 1/3, with α and β acute. What is tan(α + β)?

  7. Question 7 of 20 · Multiple Choice

    sin α = 3/5 with α in Quadrant II, and sin β = 5/13 with β in Quadrant I. What is cos(α - β)?

  8. Question 8 of 20 · Multiple Choice

    Simplify sin(x + π).

  9. Question 9 of 20 · Multiple Choice

    Simplify cos(x + 3π/2).

  10. Question 10 of 20 · Multiple Choice

    In the chord proof of cos(α - β), why is the distance from A(cos α, sin α) to B(cos β, sin β) equal to the distance from C(cos(α - β), sin(α - β)) to D(1, 0)?

  11. Question 11 of 20 · Multiple Choice

    How is sin(α + β) derived from the cosine difference formula?

  12. Question 12 of 20 · Multiple Choice

    tan α = 4 and tan β = 1/4, with α and β acute. Why can't you use the formula to find tan(α + β)?

  13. Question 13 of 20 · Multiple Choice

    Solve cos x cos(π/6) + sin x sin(π/6) = 0 for 0 ≤ x < 2π.

  14. Question 14 of 20 · Multiple Choice

    A student writes cos(60° + 30°) = cos 60° + cos 30°. Which calculation shows the error?

  15. Question 15 of 20 · Short Answer

    Use the formulas for sin(α + β) and cos(α + β) to prove that tan(α + β) = (tan α + tan β)/(1 - tan α tan β). State when the proof is valid.

  16. Question 16 of 20 · Short Answer

    sin α = -12/13 with α in Quadrant III, and cos β = 15/17 with β in Quadrant I. Find sin(α + β) and cos(α + β), and name the quadrant of α + β.

  17. Question 17 of 20 · Short Answer

    Starting from cos(α - β) = cos α cos β + sin α sin β, prove the formula for cos(α + β).

  18. Question 18 of 20 · Short Answer

    Prove the identity sin(α + β) sin(α - β) = sin²α - sin²β.

  19. Question 19 of 20 · Short Answer

    tan α = 5 and tan β = 2/3, with α and β acute. Find tan(α - β) and the angle α - β.

  20. Question 20 of 20 · Short Answer

    Solve sin x cos(π/4) - cos x sin(π/4) = 1/2 for 0 ≤ x < 2π.

0 of 20 answered · 0 correct

06

Frequently Asked Questions

10 Questions

What does HSF.TF.C.9 mean?

HSF.TF.C.9 asks students to prove the formulas for sin(α ± β), cos(α ± β) and tan(α ± β) and to use them to solve problems. For example, students prove cos(α - β) = cos α cos β + sin α sin β on the unit circle and then use cos(60° + 45°) to find the exact value of cos 105°.

Is HSF.TF.C.9 in Algebra 2 or Precalculus?

It is usually taught in Precalculus. The standard is marked (+), which Common Core uses for additional mathematics that students need for advanced courses such as calculus. Some Algebra II honors courses include the formulas without all of the proofs.

How do you remember the sum and difference formulas?

Two patterns help. Sine mixes the functions and keeps the sign: sin(α ± β) = sin α cos β ± cos α sin β. Cosine keeps the functions together and flips the sign: cos(α ± β) = cos α cos β ∓ sin α sin β. For tangent, the numerator keeps the sign and the denominator flips it. Proving them once makes them easier to rebuild.

Which formula should be proved first?

Most textbooks start with cos(α - β), because the unit circle distance proof works for any angles, not only acute ones. The other five follow from it with the even and odd properties, the cofunction identities and the definition of tangent. The rectangle proof in Diagram 2 is a good second argument, but it covers only acute angles.

Why is sin(α + β) not equal to sin α + sin β?

Because sine is not a linear function. One counterexample is enough: sin(30° + 60°) = 1, but sin 30° + sin 60° ≈ 1.37. Any value above 1 is impossible for a sine, which shows the error right away.

How do you find exact values like sin 75° or cos 105°?

Write the angle as a sum or difference of angles with known values, such as 30°, 45°, 60° and their multiples: 75° = 45° + 30° and 105° = 60° + 45°. Then apply the formula and simplify. Check the sign with the quadrant of the original angle and the decimal with a calculator.

What are common mistakes with these formulas?

Common errors include distributing sine or cosine over a sum, using a plus sign in cos(α + β), mixing up the numerator and denominator signs in the tangent formula, and using the wrong sign for cos α or sin β when the angles are in different quadrants. Naming the quadrant of each angle before computing prevents the last error.

What kinds of problems do these formulas solve?

They give exact values, simplify expressions such as cos(x + π) = -cos x, prove other identities, and turn expressions like sin x cos(π/3) + cos x sin(π/3) into a single sine so an equation can be solved. The tangent formula gives the angle between two lines from their slopes and viewing angles from heights and distances.

How do the addition formulas connect to double-angle formulas?

Setting β = α in the sum formulas gives sin 2α = 2 sin α cos α and cos 2α = cos²α - sin²α. With the Pythagorean identity, the second becomes 1 - 2sin²α or 2cos²α - 1. Double-angle formulas are usually taught right after this standard.

Where are these formulas used after Precalculus?

In calculus, they are used to find the derivatives of sine and cosine. In physics and engineering they combine waves and describe rotations: rotating the point (cos β, sin β) by an angle α gives (cos(α + β), sin(α + β)). They also explain why multiplying complex numbers in polar form adds their angles.