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SQA Higher Mathematics: Essential Formulas, Circles & Calculus

Key formulas, circle geometry, recurrence relations, differentiation, integration, and vector wave functions for SQA Higher Mathematics.

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Equation of Straight Line Given Gradient & Point

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Equation of Straight Line Given Gradient & Point

y - b = m (x - a)

, where

m

is the gradient and

(a, b)

is a point on the line. Gradient

m = tan θ

, where

θ

is angle with positive

x

-axis.

Perpendicular Lines Gradient Condition

Two lines with gradients

m_{1}

and

m_{2}

are perpendicular

⟺ m_{1} \times m_{2} = - 1

(or

m_{2} = - \frac{1}{m _{1}}

), provided neither is vertical.

Altitude, Median, and Perpendicular Bisector of Triangle

Altitude: from vertex perpendicular to opposite side. Median: from vertex to midpoint of opposite side. Perpendicular bisector: passes through midpoint of side at right angles.

Linear Recurrence Relation & Limit Condition

u_{n + 1} = a u_{n} + b

. A limit

L

exists if and only if

- 1 < a < 1

. When it exists,

L = \frac{b}{1 - a}

General Equation of Circle & Centre/Radius

x^{2} + y^{2} + 2 g x + 2 f y + c = 0

. Centre is

(- g, - f)

and radius is

r = g^{2} + f^{2} - c

, requiring

g^{2} + f^{2} - c > 0

Standard Equation of Circle with Centre

(a, b)

(x - a)^{2} + (y - b)^{2} = r^{2}

, where centre is

(a, b)

and radius is

r

Equation of Tangent to a Circle at Point

(x_{1}, y_{1})

Find radius gradient

m_{radius} = \frac{y _{1} - ( - f )}{x _{1} - ( - g )}

. Tangent gradient is

m_{tgt} = - \frac{1}{m _{radius}}

. Tangent equation:

y - y_{1} = m_{tgt} (x - x_{1})

Intersection of Line and Circle: Discriminant Rule

Substitute line

y = m x + c

into circle to get

A x^{2} + B x + C = 0

Δ = B^{2} - 4 A C

Δ > 0 ⟹

secant (2 points);

Δ = 0 ⟹

tangent (1 point);

Δ < 0 ⟹

no intersection.

Derivatives of Trigonometric Functions (Rad Angle)

\frac{d}{d x} (sin (k x)) = k cos (k x)

\frac{d}{d x} (cos (k x)) = - k sin (k x)

. Angles must be measured in radians.

Chain Rule for Differentiation

\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)

, or

\frac{d}{d x} [(a x + b)^{n}] = an (a x + b)^{n - 1}

Stationary Points & Nature Determination

Stationary points occur where

\frac{d y}{d x} = 0

. Determine nature using a nature table of signs of

\frac{d y}{d x}

(or second derivative:

f^{''} (x) > 0 ⟹

min,

f^{''} (x) < 0 ⟹

max).

Closed Interval Maximum and Minimum (Global Extrema)

Evaluate function

f (x)

at: (1) all stationary points within interval

[a, b]

, and (2) endpoints

x = a

and

x = b

. Greatest value is global max, least is global min.

Integration of Linear Composites:

\int (a x + b)^{n} d x

\int (a x + b)^{n} d x = \frac{( a x + b ) ^{n + 1}}{a ( n + 1 )} + C

, where

n \neq = - 1

and

a \neq = 0

Integration of Sine and Cosine

\int sin (k x) d x = - \frac{1}{k} cos (k x) + C

\int cos (k x) d x = \frac{1}{k} sin (k x) + C

Area Between Two Curves

Area

= \int_{a}^{b} [Upper Curve - Lower Curve] d x = \int_{a}^{b} (f (x) - g (x)) d x

, where

f (x) \geq g (x)

[a, b]

Compound Angle Identities for Sine and Cosine

sin (A \pm B) = sin A cos B \pm cos A sin B

;

cos (A \pm B) = cos A cos B \mp sin A sin B

Double Angle Formula for Sine

sin 2 A = 2 sin A cos A

Double Angle Formulas for Cosine (Three Forms)

cos 2 A = cos^{2} A - sin^{2} A = 2 cos^{2} A - 1 = 1 - 2 sin^{2} A

Wave Function:

a cos x + b sin x = k cos (x - α)

k = a^{2} + b^{2}

cos α = \frac{a}{k}

sin α = \frac{b}{k}

tan α = \frac{b}{a}

. Determine quadrant of

α

using signs of

cos α

and

sin α

3D Vectors: Magnitude and Unit Vector

∣ u ∣ = u_{1}^{2} + u_{2}^{2} + u_{3}^{2}

. Unit vector in direction of

u

\overset{u}{^} = \frac{u}{∣ u ∣}

Scalar Product and Angle Between 3D Vectors

u \cdot v = u_{1} v_{1} + u_{2} v_{2} + u_{3} v_{3} = ∣ u ∣∣ v ∣ cos θ

. Perpendicular iff

u \cdot v = 0

Collinear Points in 3D Space

Points

A, B, C

are collinear if

A B = k B C

(

k \in R

) and point

B

is common to both segments.

Section Formula (Dividing a Line in Ratio

m : n

)

Position vector of point

P

dividing

A B

in ratio

m : n

p = \frac{n a + m b}{m + n}

Logarithmic to Exponential Form Conversion

y = lo g_{a} x ⟺ x = a^{y}

(

a > 0, a \neq = 1, x > 0

). In particular,

ln x = y ⟺ x = e^{y}

Linear Law Graphs:

y = k x^{n}

y = a b^{x}

For

y = k x^{n}

: plot

lo g y

lo g x

gives straight line with gradient

n

and intercept

lo g k

. For

y = a b^{x}

: plot

lo g y

x

gives gradient

lo g b

and intercept

lo g a

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