Compound Angle Formulae
Expand sin(A ± B), cos(A ± B) and tan(A ± B), and use them for exact values of angles built from known ones and for equations mixing two angles.
What a learner can do afterwards
- Find an exact value for cos 15° by writing it as cos(45° - 30°)
- Expand sin(x + π/4) and simplify the constants
- Solve an equation after expanding its compound angle
1 · Read
Four expansions cover sums and differences. Sine keeps the middle sign, cosine flips it. Sin of A plus B is sin A cos B plus cos A sin B. Cos of A minus B is cos A cos B plus sin A sin B. Write the formula before substituting.
Cos 15 degrees splits as cos of 45 minus 30. Expand: cos 45 cos 30 plus sin 45 sin 30. Special values give root 6 plus root 2, all over 4.
Expanding helps with sums inside expressions. Sin of x plus 45 degrees is sin x cos 45 plus cos x sin 45. The constants simplify to root 2 over 2 each. Collapsing runs backwards: a sin-cos plus cos-sin pattern names one sum.
After collapsing, equations fall quickly. Sin of x plus 30 degrees equals 1 gives x plus 30 equals 90, so x is 60. Track the shifted range: x between 0 and 360 with a 30 shift solves across 30 to 390.
Split into known angles, expand with careful signs, and track the range through every shift.
More to read
9.2 Sum and Difference Identities
Exact values for awkward angles come from splitting them into special ones: cos 15° = cos(45° − 30°) = cos45 cos30 + sin45 sin30. Memorise the four expansions with their signs.
Sine keeps the middle sign, cosine flips it: sin(A+B) expands with +, cos(A−B) expands with +. Writing the formula before substituting stops sign slips.
7.2 Sum and Difference Identities
Expanding sin(x + 45°) gives sin x cos 45° + cos x sin 45°, useful when equations hide sums inside. Collapsing works backwards too: sin x cos 30° + cos x sin 30° is just sin(x + 30°), ready to solve.
Spot the pattern sin-something-cos plus cos-something-sin and name the sum. One collapsed function solves faster than two expanded ones.
7.2 Sum and Difference Identities
After collapsing, equations fall quickly: sin(x + 30°) = 1 gives x + 30° = 90°, so x = 60°. The shift rides along inside, then subtracts off at the end.
Check the range after shifting: x in 0 to 360 with a +30 shift means solving across 30 to 390. Boundary shifts hide extra solutions, so track the stretched interval like a 2x problem.
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2 · Watch
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Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
17 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.