Compound, double and triple angles, sums and products
Use the addition formulae with exact values to find sin 75°, then work with double and triple angles, turn a sum of sines into a product, and write √3sinx + cos x as a single sine.
- UK: Year 12
- Germany: grade 10
- Iran: grade 11
- Afghanistan: grade 10
- Syria: grade 11
- Ukraine: grade 10
- Russia: grade 10
Guide
For Afghan Grade 10 (chapter 5, applications of trigonometry), Iran Grade 11 and A-level. Every answer comes from the exact values for 30°, 45° and 60°; the picture shows the angles involved on the unit circle.
The addition formulae: sin (A + B) = sin AcosB + cos AsinB, cos (A + B) = cos AcosB − sin AsinB and tan (A + B) = tan A + tan B1 − tan AtanB. For A − B the signs change. Sine does not distribute over a sum: sin (A + B) is not sin A + sin B.
Putting B = A gives the double-angle formulae sin 2A = 2sinAcosA and cos 2A = cos2A − sin2A, and then sin 3A = 3sinA − 4sin3A.
A sum of sines becomes a product: sin A + sin B = 2sin(A + B2)cos (A − B2). And asinx + bcosx = Rsin(x + α) with R = √(a2 + b2) and tanα = ba.