Twelfth Grade · Math · week 22

Sum and difference identities

Use the sum and difference identities for sine, cosine, and tangent to find exact trigonometric values and simplify expressions.

The sum and difference identities let you find an exact trig value for an angle that is not one of the special angles, as long as you can write it as a sum or difference of special angles. cos(A+-B) = cos(A)cos(B) -+ sin(A)sin(B); sin(A+-B) = sin(A)cos(B) +- cos(A)sin(B); notice the sign pattern flips between the two.

The tangent version, tan(A+-B) = (tan(A)+-tan(B)) / (1 -+ tan(A)tan(B)), follows from dividing the sine identity by the cosine identity. These identities are not just algebra tricks -- they are the reason a co-function fact like sin(15)=cos(75) is guaranteed to be true, since 15 and 75 are complementary.

📋 For the teacher
Teach it (10–15 min):
  • Post all four identities (sin and cos, sum and difference) together, and point out the sign pattern flips between sine and cosine -- that single observation prevents most memory slips.
  • Always pick a decomposition using only the three special angles (30, 45, 60) plus their sums or differences -- more than one valid choice often exists, and any correct one works.
  • Use the co-function check (like sin(15)=cos(75)) as free error-checking whenever two related angles both get computed.
  • Narrate the tangent identity as built FROM the sine and cosine ones, rather than a fourth thing to memorize separately.
Talk about it:
  • Why does the sign in the cosine sum/difference identity flip compared to the sign in the original sum or difference?
  • How could you find the exact value of tan(15 degrees) using what you know?
  • Why is it useful that these identities work for ANY two angles, not just the special ones?

Mastery looks like: They select a correct sum-or-difference decomposition using special angles and correctly apply the identity to find an exact trigonometric value.

Common stumbles: Mixing up the sign pattern between the sine and cosine identities; treating sin(A+B) as if it were sin(A)+sin(B).

See it visually

Interactive, spoken-aloud tools that show this idea in motion.

SubtractionTake away and count back, step by step.try →AdditionSee groups combine — with counters, blocks, and the number line.try →Unit CircleAngles, radians, and sine & cosine come alive.try →
Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
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