LEMMA

Advanced trigonometry

Angle-sum and double-angle formulas

ALEKS placement

Angle-sum and double-angle formulas rewrite one awkward angle in terms of familiar ones.

What this covers

sin(A±B)=sinAcosB±cosAsinB,cos2A=1−2sin2A
  • Split an unfamiliar angle into a sum or difference of special angles.
  • Apply the matching formula exactly, watching the sign pattern.
  • Cosine's sum formula flips the sign; sine's keeps it.
  • Choose the double-angle version whose form matches what you need.
  • If you need a missing trig value, find its size with an identity and use the quadrant to choose its sign.

Worked example

Worked example

Find sin(75°) exactly.

  1. sin(45∘+30∘)Split into two special angles.
  2. sin45∘cos30∘+cos45∘sin30∘Apply the sine sum formula.
  3. 4√6+√2​Substitute the exact values and combine.
Another worked example

sin θ = 3/5 and θ is in quadrant II. Find cos(2θ) and sin(2θ).

  1. cosθ=−√1−259​​=−54​Use the Pythagorean identity and choose the negative cosine required by quadrant II.
  2. cos2θ=1−2sin2θ=1−2518​=257​This cosine double-angle form uses the sine value directly.
  3. sin2θ=2sinθcosθ=2⋅53​⋅(−54​)=−2524​The sine double-angle formula needs both coordinates, including cosine's negative sign.

A common mistake

A mistake Lemma catches

From cos15∘=cos(45∘−30∘)

cos(45∘−30∘)=cos45∘cos30∘−sin45∘sin30∘cos(45∘−30∘)=cos45∘cos30∘+sin45∘sin30∘

“Cosine's formula flips the sign: cos(A − B) = cos A cos B + sin A sin B. Check with values: cos 15° ≈ 0.966 and (√6 + √2)/4 ≈ 0.966, while the minus version gives (√6 − √2)/4 ≈ 0.259, which is cos 75°.”

Try one

Sample problem

Evaluate exactly.

sin(15∘)

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Where this fits

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