NATA Mathematics for Architecture Syllabus

Give the sine and cosine addition formulas.

sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B.

State the double-angle formulas for sin 2θ and cos 2θ.

sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.

Give the general solution of sin θ = sin α and cos θ = cos α.

sin θ = sin α → θ = nπ + (−1)ⁿ α; cos θ = cos α → θ = 2nπ ± α (n ∈ ℤ).

In a heights and distances problem, define angle of elevation and angle of depression.

Angle of elevation: angle above the horizontal to an object higher than the observer. Angle of depression: angle below the horizontal to an object lower than the observer.

How do you find the height of a tower given a horizontal distance d and angle of elevation θ from the base level?

Height = d · tan θ (using tan θ = opposite/adjacent in the right triangle).

State the principal value ranges of arcsin, arccos, and arctan.

arcsin x ∈ [−π/2, π/2]; arccos x ∈ [0, π]; arctan x ∈ (−π/2, π/2).

What is the relationship between arcsin x + arccos x for x in [−1, 1]?

arcsin x + arccos x = π/2.

State the Law of Sines for a triangle with sides a, b, c and opposite angles A, B, C.

a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius.

State the Law of Cosines.

a² = b² + c² − 2bc cos A (and cyclic variants for the other sides/angles).

Give two formulas for the area of a triangle using its sides/angles.

Area = (1/2)ab sin C; and Heron's formula: Area = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2.

Give the distance formula and the section formula (internal division) for two points in the plane.

Distance = √[(x₂−x₁)² + (y₂−y₁)²]. Section formula (ratio m:n): ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)).

State the slope formula and the slope–intercept form of a straight line.

Slope m = (y₂−y₁)/(x₂−x₁); slope–intercept form: y = mx + c.

What is the condition for two lines to be parallel and for them to be perpendicular (in terms of slopes)?

Parallel: m₁ = m₂. Perpendicular: m₁ · m₂ = −1.

Give the distance from a point (x₀, y₀) to the line ax + by + c = 0.

Distance = |ax₀ + by₀ + c| / √(a² + b²).

State the standard equation of a circle with centre (h, k) and radius r, and the general form.

Standard: (x − h)² + (y − k)² = r². General: x² + y² + 2gx + 2fy + c = 0, with centre (−g, −f) and radius √(g² + f² − c).

What is a locus, and what is the typical method to find its equation?

A locus is the set of all points satisfying a given geometric condition. Method: let the moving point be (x, y), translate the condition into an equation in x and y, then simplify.

Give the standard equation of a parabola y² = 4ax and its focus and directrix.

y² = 4ax opens rightward; focus (a, 0); directrix x = −a; vertex at origin.

State the standard equation of an ellipse (a > b) and its eccentricity.

x²/a² + y²/b² = 1; eccentricity e = √(1 − b²/a²), with 0 < e < 1; foci at (±ae, 0).

State the standard equation of a hyperbola and its eccentricity.

x²/a² − y²/b² = 1; eccentricity e = √(1 + b²/a²), with e > 1; foci at (±ae, 0).

Compare the eccentricities of a circle, parabola, ellipse, and hyperbola.

Circle: e = 0; ellipse: 0 < e < 1; parabola: e = 1; hyperbola: e > 1.

Give the distance formula and midpoint formula for two points in 3D space.

Distance = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]. Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2).

Define direction cosines of a line in 3D and the relation they satisfy.

Direction cosines l, m, n are the cosines of the angles the line makes with the x, y, z axes; they satisfy l² + m² + n² = 1.

State the formal definition of the derivative of f(x) (first principles).

f'(x) = lim(h→0) [f(x + h) − f(x)] / h, when the limit exists.

State the conditions for a function to be continuous at a point x = a.

f(a) is defined, lim(x→a) f(x) exists, and lim(x→a) f(x) = f(a). (Differentiability at a implies continuity at a, but not vice versa.)

State the product rule and quotient rule for differentiation.

Product: (uv)' = u'v + uv'. Quotient: (u/v)' = (u'v − uv')/v².

How do you use the first and second derivative to classify a critical point as a maximum or minimum?

At f'(x) = 0: if f''(x) < 0 it is a local maximum; if f''(x) > 0 it is a local minimum; if f''(x) = 0 the test is inconclusive.

State the power rule for indefinite integration and the integral of 1/x.

∫xⁿ dx = x^(n+1)/(n+1) + C (n ≠ −1); ∫(1/x) dx = ln|x| + C.

State the Fundamental Theorem of Calculus for a definite integral.

∫ from a to b of f(x) dx = F(b) − F(a), where F is an antiderivative of f (F' = f).

How do you find the area enclosed between a curve y = f(x) and the x-axis from x = a to x = b (with f ≥ 0)?

Area = ∫ from a to b of f(x) dx. If f can be negative, integrate |f(x)| or split at the roots.

Define mean, median, and mode as measures of central tendency.

Mean: arithmetic average (Σx/n). Median: middle value of ordered data. Mode: most frequently occurring value.

Give the formulas for variance and standard deviation of a data set.

Variance σ² = Σ(xᵢ − x̄)²/n; standard deviation σ = √(variance).

State the classical (theoretical) definition of probability and its range.

P(E) = (number of favourable outcomes)/(total number of equally likely outcomes); 0 ≤ P(E) ≤ 1, and P(E) + P(not E) = 1.

Give the area and perimeter (circumference) of a circle, and the area of a triangle and rectangle.

Circle: area = πr², circumference = 2πr. Triangle: area = (1/2)·base·height. Rectangle: area = length × breadth, perimeter = 2(l + b).

Give the area and perimeter of a trapezium and the area of a parallelogram.

Trapezium area = (1/2)(sum of parallel sides)·height. Parallelogram area = base × height.

Give the surface area and volume formulas for a sphere.

Surface area = 4πr²; volume = (4/3)πr³.

Give the curved surface area, total surface area, and volume of a right circular cylinder.

CSA = 2πrh; TSA = 2πr(h + r); volume = πr²h.

Give the curved surface area, total surface area, and volume of a right circular cone.

CSA = πrl (slant height l = √(r² + h²)); TSA = πr(l + r); volume = (1/3)πr²h.

Give the total surface area and volume of a cube of side a, and of a cuboid with dimensions l × b × h.

Cube: TSA = 6a², volume = a³. Cuboid: TSA = 2(lb + bh + hl), volume = l·b·h.