Compound, double and triple angles, sums and products
Use the addition formulae with exact values to find sin ٧٥°, then work with double and triple angles, turn a sum of sines into a product, and write √٣sin x + cos x as a single sine.
- سوريا: الصف ١١
- إيران: الصف ١١
- أفغانستان: الصف ١٠
- أوكرانيا: الصف ١٠
- روسيا: الصف ١٠
- ألمانيا: الصف ١٠
- بريطانيا: السنة ١٢
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الدليل
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 B١ − 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 ٢A = ٢sin AcosA and cos ٢A = cos٢A − sin٢A, and then sin ٣A = ٣sin A − ٤sin٣A.
A sum of sines becomes a product: sin A + sin B = ٢sin (A + B٢)cos (A − B٢). And asinx + bcosx = Rsin(x + α) with R = √(a٢ + b٢) and tanα = ba.