1
Sine keeps the sign; cosine flips it.
sin(A ± B)
= sin A cos B ± cos A sin B
same sign
cos(A ± B)
= cos A cos B ∓ sin A sin B
OPPOSITE sign
tan(A ± B)
= tan A ± tan B1 ∓ tan A tan B
both signs appear
The cosine sign is the one that catches everyone
cos(A + B) has a minus and cos(A − B) has a plus. Every other formula keeps the sign
you were given. Check the cosine one against a known value before trusting it:
cos(90° + 0°) should give 0, and cos 90 cos 0 − sin 90 sin 0 = 0 ✓ The same test kills the tempting shortcut
cos(A + B) = cos A + cos B: with A = B = 45°, cos 90° = 0, but cos 45° + cos 45° = √2.
One known angle is enough to expose a wrong formula.
2
Sine and cosine mix; they do not stay separate.
sin(A + B)
both terms mix sin and cos
sin·cos + cos·sin
cos(A + B)
terms stay pure
cos·cos − sin·sin
the memory hook
"sine mixes, cosine matches"
and cosine flips the sign
Getting the pattern of the terms right is half the work; getting the sign right is the
other half. Both are worth a ten-second check with a known angle.
3
Build an awkward angle from two familiar ones.
105° = 60° + 45°
both are special angles
cos(60 + 45) = cos60 cos45 − sin60 sin45
plus inside, MINUS in the formula
= (12)(√22) − (√32)(√22)
substitute the known values
= √2 − √64
about −0.259 ✓
Check the size: 105° is just past 90°, so its cosine should be small and negative. An
answer of −0.259 passes; a positive answer means the cosine sign was not flipped. Any sum or difference of 30°, 45°, 60° and 90° works the same way: 75° = 45° + 30°, and 15° = 45° − 30°, so cos 15° = cos 45° cos 30° + sin 45° sin 30°.
4
Running the formula backwards is the real skill.
sin80 cos50 − cos80 sin50
matches sin A cos B − cos A sin B
= sin(80 − 50)
A = 80, B = 50
= sin 30° = 12
an exact value from two that are not
Four unpleasant values collapse into one exact answer. Spotting the pattern in a long
expression is worth far more marks than expanding one, and exam questions know it.
5
The cosine pattern backwards: the sign between the terms decides.
"cos·cos" and "sin·sin" side by side is a cosine formula. The sign between them tells us whether the angles add or subtract, the opposite way round.
Simplify cos 50° cos 20° + sin 50° sin 20°.
cos50 cos20 + sin50 sin20
pure terms: a cosine formula
PLUS between the terms, so it is cos(A − B)
cosine flips the sign
= cos(50 − 20) = cos 30°
A = 50, B = 20
Adding the angles because the sign is plus
Reading the + as "add" gives cos 70°, which is not a special angle at all — a sign that the pattern was misread. For cosine, + between the terms means subtract the angles, and − means add them.
6
Variations: a quadrant II angle, tangent, and radians.
The same formulas, with a negative piece, a fraction to tidy, and π in place of degrees.
A quadrant II angle: sin A = 513 (A in II), cos B = 35 (B in I). Find cos(A − B).
cos A = −1213, sin B = 45
cosine is negative in II
cos(A − B) = cos A cos B + sin A sin B
minus inside, plus outside
= (−1213)(35) + (513)(45) = −3665 + 2065
substitute
= −1665
a negative answer is fine
Using +1213 gives 5665. The quadrant sign has to be fixed before substituting.
Tangent: tan 75° = tan(45° + 30°)
= 1 + √331 − √33
tan 45 = 1, tan 30 = √33
= 3 + √33 − √3
multiply top and bottom by 3
= 9 + 6√3 + 39 − 3 = 12 + 6√36
multiply by 3 + √3 top and bottom
Check: 75° is steep, so its tangent should be well above 1.
Radians: sin(π12) = sin(π3 − π4)
= sin(π3)cos(π4) − cos(π3)sin(π4)
sine keeps the minus
= (√32)(√22) − (12)(√22)
π3 is 60°, π4 is 45°
π12 is 15°, so its sine should be small and positive. It is.
7
You usually have to find the missing pieces first.
sin A = 817 → cos A = 1517
8-15-17 triangle, quadrant I
cos B = 45 → sin B = 35
3-4-5 triangle
sin(A+B) = (817)(45) + (1517)(35)
sin mixes, sign unchanged
= 3285 + 4585 = 7785
just under 1, which is plausible
cos(A + B) = cos A cos B − sin A sin B
The quadrant matters: in quadrant I both missing values are positive, but in quadrant
II cos A would have been −1517 and the answer would change completely. Always use the
Pythagorean identity for the size and the quadrant for the sign.
Remember the flow
Sine mixes and keeps the sign · Cosine matches and flips it · Split awkward angles into special ones · Watch for the pattern in reverse
Test any formula you are unsure of against a known angle before using it.
On the exam
Tests ask for an exact value such as cos 75° or sin 15°, for a long expression to be collapsed back into one function, or for sin(A + B) given two ratios and quadrants. The fastest route: sine mixes and keeps the sign, cosine matches and flips it; split the angle into 30°, 45°, 60° pieces, or read A and B straight off the pattern. A 10-second check: estimate the angle's sine or cosine on the unit circle and see the exact answer has the right size and sign.
