MODULE 2tCHAPTER 9
SUMMARY
Trigonometry
Graphs and general solutions Trigonometric identities
Graph of y = sin x sin2 θ + cos2 θ = 1
y
1 tan2 θ + 1 = sec2 θ
1 + cot2θ = cosec2θ
x
0 π π 3π 2π
2 2
–1
sin (A ± B) = sin A cos B ± cos A sin B
cos (A ± B) = cos A cos B ∓ sin A sin B
tan (A ± B) = tan A ± tan B
– tan A tan B
Graph of y = cos x 1+
y
1
x sin 2θ = 2 sin θ cos θ
0 π π 3π 2π cos 2θ = cos2θ – sin2θ
2 2
cos 2θ = 2 cos2θ – 1
–1 cos 2θ = 1 – 2 sin2θ
tan 2θ = 2 tan θ2
1 – tan θ
Graph of y = tan x
y
1
If t = tan θ , then:
x 2
π π
sin θ = 2 2t
– 0
2 2
t +1
–1
cos θ = 1 – t 2
2
1+t
tan θ = 2t 2
Graph of y = cosec x 1–t
y
1
x
2sin A cos B = sin (A + B) + sin (A – B)
0 π 2π 2cos A sin B = sin (A + B) – sin (A – B)
–1
2cos A cos B = cos (A + B) + cos (A – B)
–2sin A sin B = cos (A + B) – cos (A – B)
Graph of y = sec x sin A + sin B = 2sin A + B cos A – B
y
2 2
sin A – sin B = 2cos A + B sin A – B
2 2
cos A + cos B = 2cos A + B cos A – B
1
x
0 π π 3π 2 2
cos A – cos B = –2 sin A + B sin A – B
–1 2 2
2 2
a cos θ + b sin θ ≡ r cos θ
Graph of y = cot x where r > 0 and 0° < α < 90°
y
x Maximum (a cos θ + b sin θ ) = r,
–π 0 π
when cos (θ – α) = 1
Minimum (a cos θ + b sin θ ) = –r,
General solution of sin x: when cos(θ – α) = –1
x = nπ + (–1)nα, n ∈
General solution of cos x:
x = 2nπ ± α, n ∈
General solution of tan x:
x = nπ + α, n ∈
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MODULE 2tCHAPTER 10
SUMMARY
Coordinate geometry
Circles Ellipses Parabolas
A circle is the locus of a point An ellispse is the locus of points, A parabola is a set of all points in a plane, which
which moves in a plane so that the sum of whose distance from two are at the same distance from a fixed point P as
it is equidistant from a fixed fixed points is constant. The two fixed they are from a fixed line l. P is called the focus
point. points are the foci of the ellipse. and l is called the directrix.
Equation of a circle centre (0, 0) Equation of an ellipse with centre (0, 0) Equation of a parabola with centre
and radius r : and foci at (c, 0) and (–c, 0) is: (0, 0) and opening upwards is:
x2 + y2 = r2 x2 + y2 = 1 x2 = 4ay
a2 b2
Equation of a circle centre (a, b) Equation of a parabola with centre
and radius r : Equation of an ellipse with foci (0, c) (0, 0) and opening to the right is:
(x – a)2 + (y – b)2 = r2 and (0, –c) is: y2 = 4ax
x2 + y2 = 1
a2 b2
General equation of a circle: Equation of a parabola with axis of symmetry
Ax2 + By2 + Cx + Dy + E = 0, A = B Equation of an ellipse with centre parallel to the y-axis is:
(h, k) and major axis parallel to the y = ax2 + bx + c a ≠ 0
x-axis is:
(x – h)2 (y – k)2 Equation of a parabola with axis of symmetry
+ =1
(a, b) a2 b2 parallel to the x-axis is:
r Foci are at (h + c, k) and (h – c, k). x = ay2 + by + c a ≠ 0
Equation of an ellipse with centre Equation of a parabola in vertex form:
Parametric equation of a circle (h, k) and major axis parallel to the y = a(x – h)2 + k
centre (a, b) and radius r : y-axis is: or y = a(y – k)2 + h
x = a + r cos θ y = b + r sinθ (x – h)2 (y – k)2
+ =1
or x = a + r sinθ y = b + r cosθ b2 a2 Equation of a parabola in conics form:
Foci are at (h, k + c) and (h, k – c).
4p(y – k) = (x – h)2
Gradient of the normal at or 4p(x – h) = (y – k)2
y –b
P= 0 General equation of an ellipse:
x0 – a
Ax2 + B2 + Cx + Dy + E = 0, A ≠ B where p is the distance from the vertex
to the focus (also the distance from the
Graph of an ellipse centre C, vertex to the directrix).
P foci F1 and F2 (2p is the distance from the focus to
(a, b) (x0 , y0)
r the directrix).
Minor axis
Axis of symmetry
Since the tangent is perpendicular Major
V2 Centre V1 P
to the radius, the gradient of the F1 F2 axis d
y –b d F
tangent of P is –( 0 ) a
x0 – a a
V
Dir
Given three points on a ect
rix
circle we can find its equation ,D
Parametric equations of an ellipse centre
by using the general form
(h, k) and radius r :
(x – a)2 + (y – b)2 = r2.
x = h + a cos t y = k + b sin t
We form three equations Parametric equations of a parabola are:
and solve them simultaneously x = at2 y = 2at t ∈ a>0
to find a, b and r. Gradient of the tangent at
P(a cos t, b sin t) is
Gradient of the tangent to a parabola
Or, if the circle passes –b cos t 1
through P, Q and R, the a sin t at P(at2, 2at) is t
centre of the circle is the and the gradient of the normal is
point of intersection of the a sin t
Gradient of the normal to a parabola at
perpendicular bisectors b cos t
P(at2, 2at)) is – t.
of PQ and QR. The radius
can be found using the
centre and any of the Equation of the tangent can be found
points P, Q and R. by using the gradient and point.
299
M O DUL E 2
SUMMARY
Vectors in three dimensions ( 3)
Vectors Lines Planes
u1 v1 Let a be a point on a line and m be Let nˆ be a unit vector perpendicular to a
Let u = u2 and v = v2 the direction vector. The equation of plane and d be the distance from the
u3 v3 origin to the plane. The equation of the
the line is
plane is:
r = a + λm, λ ∈ r ⋅ nˆ = d
u1 + v1
u2 + v2 Let p and q be two points on a line.
u+v=
u3 + v3 The equation of the line is
λ λ Let a be a point on a plane and n a vector
perpendicular to the plane. The equation
of the plane is:
u1 – v1 Vector equation of a line: r⋅n=a⋅n
u−v= u2 – v2 a x0
u3 – v3 r= b +λ y0
c z0 Vector equation of a plane:
A
u1 λu1 r. B ∙D
Cartesian equation of a line:
λu2 C
λ u2 = , λ∈ x–a y–b z–c
= =
u3 λu3 x0 y0 z0
Cartesian equation of a plane:
u = v iff u1 = v1, u2 = v2, u3 = v3 Parametric equations of a line:
Ax + By + Cz = D
x = a + λ x0
A
u · v = u1v1 + u2v2 + u3v3 y = b +λ y0 λ ∈
where B is a vector perpendicular to
z = c + λ z0 C
u = u12 + u22 + u32 the plane.
Skew lines are lines that are not
A unit vector is a vector with
parallel and do not intersect.
magnitude 1 unit.
The angle between two lines is the
A unit vector in the direction of u is angle between their direction vectors.
u .
|u|
Two lines are perpendicular iff the
u · v = |u||v| cos θ , where θ is the scalar product of their direction
angle between u and v. vectors is 0.
u and v are perpendicular iff u · v = 0.
u and v are parallel iff one is a scalar
multiple of the other.
1 0 0
i= 0 , j= 1 , k= 0
0 0 1
a
b = ai + bj + ck
c
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