Worked examples: HSC Maths Extension 1 Formula Sheet

Formula index with worked examples

1. Polynomials · cubic roots and coefficients

   Formula or reference: For ax³ + bx² + cx + d = 0 with roots α, β, γ: α + β + γ = −b/a, αβ + αγ + βγ = c/a and αβγ = −d/a.

   Units and conditions: a ≠ 0. Count roots with multiplicity.

   Worked example: The roots 1, 2 and 3 give sum 6, pairwise sum 11 and product 6, so the corresponding monic polynomial is x³ − 6x² + 11x − 6.

2. Trigonometry · compound angles

   Formula or reference: 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)/(1 − tan A tan B).

   Units and conditions: Use consistent angle units. For the tangent form the individual tangents and denominator must be defined.

   Worked example: With A = 45° and B = 30°, the sine and cosine of 75° are (√6 + √2)/4 and (√6 − √2)/4, and its tangent is 2 + √3.

3. Trigonometry · half-angle substitution

   Formula or reference: For t = tan(A/2): sin A = 2t/(1+t²), cos A = (1−t²)/(1+t²), tan A = 2t/(1−t²).

   Units and conditions: Use where t is defined, and t ≠ ±1 for the tangent form.

   Worked example: At t = 1/2, the sine, cosine and tangent of A are 4/5, 3/5 and 4/3.

4. Trigonometry · products to sums

   Formula or reference: cos A cos B = [cos(A−B)+cos(A+B)]/2. sin A sin B = [cos(A−B)−cos(A+B)]/2. sin A cos B = [sin(A+B)+sin(A−B)]/2. cos A sin B = [sin(A+B)−sin(A−B)]/2.

   Units and conditions: Keep A and B in the same angle unit, and preserve their order in the difference.

   Worked example: With A = 60° and B = 30°, the four products in order are √3/4, √3/4, 3/4 and 1/4.

5. Trigonometry · squared sine and cosine

   Formula or reference: sin²(nx) = [1 − cos(2nx)]/2. cos²(nx) = [1 + cos(2nx)]/2.

   Units and conditions: Use consistent angle units.

   Worked example: For nx = π/3, cos(2nx) = −1/2, so the squared sine is 3/4 and the squared cosine is 1/4.

6. Calculus · derivatives of inverse trig functions

   Formula or reference: d(arcsin f)/dx = f′/√(1−f²). d(arccos f)/dx = −f′/√(1−f²). d(arctan f)/dx = f′/(1+f²).

   Units and conditions: For the first two derivatives use |f| < 1. Inverse trig angles are in radians.

   Worked example: For f(x) = 2x, the three derivatives at x = 0 are 2, −2 and 2.

7. Calculus · inverse-trig antiderivatives

   Formula or reference: ∫ f′/√(a²−f²) dx = arcsin(f/a) + C. ∫ f′/(a²+f²) dx = arctan(f/a)/a + C.

   Units and conditions: Use a > 0. The first form applies where |f| < a.

   Worked example: For f(x) = 2x and a = 3, antiderivatives of 2/√(9−4x²) and 2/(9+4x²) are arcsin(2x/3) + C and arctan(2x/3)/3 + C.

8. Combinatorics · permutations and combinations

   Formula or reference: P(n,r) = n!/(n−r)!. C(n,r) = n!/[r!(n−r)!].

   Units and conditions: Use integer 0 ≤ r ≤ n. Permutations count order; combinations do not.

   Worked example: From 5 people there are P(5,2) = 20 ways to fill two different roles and C(5,2) = 10 ways to select an unordered pair.

9. Algebra · binomial expansion

   Formula or reference: The term with aʳ in (x+a)ⁿ is C(n,r)x^(n−r)aʳ, for r = 0, …, n.

   Units and conditions: n is a non-negative integer.

   Worked example: The term with a² in (x+a)⁵ is 10x³a².

10. Probability · binomial distribution

    Formula or reference: For X ~ Bin(n,p), P(X=r) = C(n,r)pʳ(1−p)^(n−r), E(X) = np and Var(X) = np(1−p).

    Units and conditions: Use n independent trials with fixed success probability p.

    Worked example: For four trials with p = 1/2, P(X=2) = 6/16, E(X) = 2 and Var(X) = 1.

11. Vectors · magnitude and scalar product

    Formula or reference: |(x,y)| = √(x²+y²). a · b = a₁b₁ + a₂b₂ = |a||b| cos θ.

    Units and conditions: To find an angle, neither vector can be zero.

    Worked example: For a = (2,0) and b = (1,√3), the dot product is 2 and both magnitudes are 2, giving θ = 60°.

12. Vectors · line equation

    Formula or reference: r = a + λb

    Units and conditions: a is a position vector on the line, b is a non-zero direction vector, and λ is real.

    Worked example: For a = (1,2) and b = (3,−1), λ = 2 gives the point (7,0).

More worked examples

Mathematics Extension 1 uses NESA's combined Advanced and Extension reference sheet. These original examples practise three different decisions: selecting an angle relation, locating a binomial term and separating a differential equation.

1. Angle between vectors

   Relationship: a · b = |a||b| cos θ for non-zero vectors.

   Find the angle between a = (1,0) and b = (1,1).

   Show worked response and error check

   a · b = 1, |a| = 1 and |b| = √2. Thus cos θ = 1/√2 and θ = 45°. The dot product is a scalar, not a vector.

   Check: Divide by both magnitudes; the dot product alone is not cos θ.

2. A binomial coefficient

   Relationship: The term selecting k factors of bx from (a + bx)^n is C(n,k)a^(n−k)(bx)^k.

   Find the coefficient of x² in (1 + 2x)^5.

   Show worked response and error check

   Choose k = 2. The term is C(5,2) × 1^3 × (2x)^2 = 10 × 4x² = 40x². The coefficient requested is 40.

   Check: The factor 2 is also squared; C(5,2) alone is only the number of selections.

3. An initial-value model

   Relationship: Separate dy/dx = ky as dy/y = k dx when y is non-zero.

   Solve dy/dx = 2y with y(0) = 3.

   Show worked response and error check

   Integrating gives ln|y| = 2x + C, so y = Ae^(2x). The initial condition makes A = 3, hence y = 3e^(2x). Substitution into the original equation verifies the solution.

   Check: The constant is fixed by the initial condition. Differentiating the final expression is a quick independent check.