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. Splitting turned an unknown angle into known ones.
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.
2 · Watch
Take it off screen
Where it sits
This opens up
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 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.