Picture: open the angle into two known pieces: You can find sin and cos of nice angles like 30°, 45°, 60° exactly. But what about 75°? It is not one of the standard angles — yet 75° = 45° + 30°, both of which you DO know.
The compound-angle formulas let you trade the awkward single angle for two friendly ones. They are in the formula booklet, so you don't memorise them — but you must know how to USE them.
The two cos forms are the tricky ones: cos(A+B) uses a minus in the middle and cos(A−B) uses a plus — the sign FLIPS compared with the bracket.
IB-style question — exact value of sin 75°
A student needs an exact (non-decimal) value of sin 75° for a Paper 1 question.
Find the exact value of sin 75°.
Step by step
- Split 75° into two angles you know exactly.
- Apply sin(A+B) = sin A cos B + cos A sin B with A = 45°, B = 30°.
- Substitute the exact surd values.
- Combine over the common denominator 4.
Final answer
sin 75° = (√6 + √2)/4 ≈ 0.966 (and the decimal check confirms it).
IB-style question — cos of a difference
Without a calculator, find the exact value of cos 15°.
Step by step
- Write 15° as a difference of known angles.
- cos(A−B) = cos A cos B + sin A sin B (the middle sign flips to +).
- Substitute the surds.
Final answer
cos 15° = (√6 + √2)/4. (Notice it equals sin 75° — because cos 15° = sin(90°−15°) = sin 75°.)
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tan adds 'over' a correction term: Dividing sin(A±B) by cos(A±B) and tidying up gives the tan formula. The top is just the tangents added/subtracted; the bottom is 1 minus/plus the product of the tangents.
Watch the signs: the bottom sign is the opposite of the top sign.
Now comes a free bonus. Put B = A in tan(A+B) and you get the double-angle formula for tan — no new rule to learn, it falls straight out.
IB-style question — derive tan 2A
Show how the double-angle formula for tan comes from the compound-angle formula.
Starting from tan(A + B), derive an expression for tan 2A.
Step by step
- Write 2A as A + A, then use tan(A+B) with B = A.
- Add the like terms on top; the product on the bottom is tan²A.
Final answer
tan 2A = 2 tan A / (1 − tan²A), obtained by putting B = A in tan(A+B).
IB-style question — exact value of tan 75°
Find the exact value of tan 75°, giving your answer in the form a + b√3.
Step by step
- Use 75° = 45° + 30° in tan(A+B).
- Substitute tan 45° = 1 and tan 30° = 1/√3.
- Rationalise by multiplying top and bottom by (√3 + 1).
- Simplify.
Final answer
tan 75° = 2 + √3 ≈ 3.73.