Core 3 Algebra Revision Notes Page 2
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1
2
3Trigonometry
Learn the formulae
1
1
1
cos
x
2
2
sec x =
cosec x =
cot x =
=
1 + tan
x = sec
x
cos
x
sin
x
tan
x
sin
x
2
2
2
2
1 + cot
x = cosec
x
sin2A = 2sinAcosA
cos2A = cos
A - sin
A
2
tan
A
2
2
tan2A =
1 + cos2A = 2cos
A
1 – cos2A = 2sin
A
2
− − − −
1
tan
A
The formulae below are in the booklet you get in the exam so no need to learn:
sin (A±B) = sinAcosB ± sinBcosA
cos(A±B) = cosAcosB
sinAsinB
m
± ± ± ±
tanA
tanB
tan (A±B) =
1 m
tan
A
tan
B
2
2
a sinx ± b cos x can be written as R sin (x ± α) or R cos (x ± α) where R = √(a
+ b
) and
the angle α is found by using the addition formula for sin(A±B) or cos (A
B).
m
Modulus
Always positive. |x| = x and |-x| = x. Graph of y=|f(x)| will not go below the x-axis. It is
y=f(x) but with the negative y bits being reflected in the x-axis.
If solving modulus questions, square both sides first then solve the quadratic.
Solving equations using the sign change rule
When the curve crosses the x-axis, f(x) will change from negative to positive or vice
versa. Use a decimal search to get the solution to the right degree of accuracy.
Use upper and lower bounds to test that the root is correct to that degree of accuracy.
Can use differentiation to find the turning points and plot the graph, then see what
values of x the root lies between.
Iteration
Rearrange equation to give x
=f(x
)… then sub in x-values for x
to give x
and sub in
r+1
r
r
r+1
again. Repeat into the solution converges to a limit. If doesn’t converge, rearrange the
equation the other way round and try again.
Volumes of Revolution
f(x) gives the radius (y-value) so the area is π(f(x))
2
and the volume is the integral of this
between the 2 limits (x=a and x=b). (π can go outside the integral sign as a constant.)
b
∫
2
π
(f(x))
dx
a
If the volume is between the curve, y-axis and y=a and y=b, then rearrange to give f(y)
y
3
1/3
e.g If y=2x
then x=(
)
so this is f(y).
2
If you want the volume of a region between 2 curves, you can calculate the volumes
separately then subtract or combine the equations first (f(x)- g(x)) then integrate this
result.
Simpson’s Rule
Used to find an approximate answer to the area under a curve if you can’t integrate it.
Approximates a curve to a quadratic needing 3 points on the curve to solve.
h=width of each interval
(Must have an even number of strips.)
b
∫
( f
) x
dx
=
1
h(y
+ 4y
+ 2y
+ 4y
+ 2y
+ …+ y
) =
1
h (y
+ 4y
+ 2y
)
0
1
2
3
4
2n
ends
odds
evens
3
3
a
Transformations
Replace x with x – k then it’s a translation of k units in the x-direction.
Replace y with y – k then it’s a translation of k units in the y-direction.
x then it’s a stretch of factor k in the x-direction.
Replace x with
k
y
Replace y with
then it’s a stretch of factor k in the y-direction.
k
Replace x with -x then it’s a reflection in the y-axis.
(Remember IT’S THE OPPOSITE
x is really a k times stretch in x)
Replace y with -y then it’s a reflection in the x-axis.
k
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