Twelfth Grade · Math · week 23

Double-angle and half-angle identities, and verifying identities

Apply the double-angle and half-angle identities, and verify trigonometric identities using a clear strategy.

The double-angle identities come from setting B=A in the sum identities: sin(2A)=2sin(A)cos(A); cos(2A)=cos^2(A)-sin^2(A), which can also be written 2cos^2(A)-1 or 1-2sin^2(A) (three equivalent forms, useful in different situations); tan(2A)=2tan(A)/(1-tan^2(A)).

The half-angle identities go the other direction, expressing a trig function of A/2 using A itself: sin(A/2)=+-sqrt((1-cos(A))/2), cos(A/2)=+-sqrt((1+cos(A))/2). The +- sign is not optional bookkeeping -- you must decide it by checking which quadrant A/2 actually falls in.

Verifying an identity means proving both sides describe the exact same function, usually by starting from the more complicated side and working it, one legal identity at a time, until it matches the simpler side. Never move terms across the equals sign as if it were an equation to solve -- an identity is proven by transforming one side, not by balancing both.

📋 For the teacher
Teach it (10–15 min):
  • Point out that all three forms of cos(2A) are equivalent, and picking the right one (in terms of only sine, or only cosine) is a strategy choice, not a memorization burden -- it depends what you already know.
  • When a half-angle's sign is ambiguous, always determine the quadrant of A/2 FIRST, before touching the square root.
  • For verifying identities, insist on transforming only ONE side at a time -- touching both sides at once invites accidentally assuming what you are trying to prove.
  • Keep a running toolbox of legal moves for verifying identities: rewrite in sin/cos, use a Pythagorean identity, combine fractions -- name which tool is being used at each step.
Talk about it:
  • Why are there three different-looking but equivalent forms of the cos(2A) identity?
  • Why must you determine the quadrant of A/2 before choosing the sign in a half-angle formula?
  • Why is it invalid, when verifying an identity, to do the same operation to "both sides" the way you would solving an equation?

Mastery looks like: They correctly apply double-angle and half-angle identities given one trig value and a quadrant, and verify an identity by transforming one side into the other.

Common stumbles: Choosing the wrong sign in a half-angle formula without checking the quadrant; manipulating both sides of an identity at once instead of transforming one side only.

See it visually

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

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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