PYQ Vault
EbooksTutor

MH HSC 12 Maths

Trigonometric Functions

Each concept is a short lesson, then questions until you get three right.

0/15

15 concepts, a short lesson each

  • Trigonometric equations0/5
  • Polar coordinates0/1
  • Solution of a triangle0/5
  • Inverse trigonometric functions0/4

Start here

Principal solutions

The solutions with 0≤θ<2π0 \le \theta < 2\pi: find the reference angle, then the quadrants where the ratio has the given sign

Start

Trigonometric equations0 of 5

  1. Principal solutionsThe solutions with 0≤θ<2π0 \le \theta < 2\pi: find the reference angle, then the quadrants where the ratio has the given signStart
  2. General solution of sin θ = sin αθ=nπ+(−1)nα\theta = n\pi + (-1)^n\alpha, n∈Zn \in \mathbb{Z}; no solution when ∣sin⁡θ∣|\sin\theta| would exceed 11Start
  3. General solutions of cos θ and tan θcos⁡θ=cos⁡α\cos\theta = \cos\alpha: θ=2nπ±α\theta = 2n\pi \pm \alpha; tan⁡θ=tan⁡α\tan\theta = \tan\alpha: θ=nπ+α\theta = n\pi + \alphaStart
  4. Equations in sin²θ, cos²θ, tan²θsin⁡2θ=sin⁡2α\sin^2\theta = \sin^2\alpha (likewise cos⁡2\cos^2, tan⁡2\tan^2) gives θ=nπ±α\theta = n\pi \pm \alpha; both signs of the root countStart
  5. Equations that reduce to the standard formsFactorise, use identities or divide out a common ratio to reach sin⁡θ=k\sin\theta = k, cos⁡θ=k\cos\theta = k or tan⁡θ=k\tan\theta = k; don't divide by a factor that can be 00Start

Polar coordinates0 of 1

  1. Polar coordinatesx=rcos⁡θx = r\cos\theta, y=rsin⁡θy = r\sin\theta; back again with r=x2+y2r = \sqrt{x^2 + y^2} and θ\theta in the point's quadrantStart

Solution of a triangle0 of 5

  1. The sine ruleasin⁡A=bsin⁡B=csin⁡C=2R\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} = 2R; the sides are in the ratio of the sines, not of the anglesStart
  2. The cosine rulecos⁡A=b2+c2−a22bc\cos A = \dfrac{b^2 + c^2 - a^2}{2bc}; the largest angle is obtuse when a2>b2+c2a^2 > b^2 + c^2Start
  3. The projection rulea=bcos⁡C+ccos⁡Ba = b\cos C + c\cos B: a side is the sum of the other two sides' projections on itStart
  4. Half-angle formulae and areaWith s=a+b+c2s = \dfrac{a + b + c}{2}: sin⁡A2=(s−b)(s−c)bc\sin\dfrac{A}{2} = \sqrt{\dfrac{(s - b)(s - c)}{bc}} and area =s(s−a)(s−b)(s−c)= \sqrt{s(s - a)(s - b)(s - c)}Start
  5. Proving triangle identitiesTurn every side into a sine (or every cosine into sides), then use the angle sum A + B + C = πStart

Inverse trigonometric functions0 of 4

  1. Principal values of inverse functionssin⁡−1\sin^{-1} in [−π2,π2][-\frac{\pi}{2}, \frac{\pi}{2}], cos⁡−1\cos^{-1} in [0,π][0, \pi], tan⁡−1\tan^{-1} in (−π2,π2)(-\frac{\pi}{2}, \frac{\pi}{2}): one angle, in that rangeStart
  2. sin⁻¹(sin x) and its relativessin⁡−1(sin⁡x)=x\sin^{-1}(\sin x) = x only for xx in the principal range; otherwise find the angle in the range with the same sineStart
  3. Properties of inverse functionssin⁡−1x+cos⁡−1x=π2\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}; tan⁡−1x+tan⁡−1y=tan⁡−1x+y1−xy\tan^{-1}x + \tan^{-1}y = \tan^{-1}\dfrac{x + y}{1 - xy} when xy<1xy < 1Start
  4. Equations in inverse functionsCombine the inverse terms with the properties, take the ratio of both sides, then keep only roots in the principal rangesStart

Practise Trigonometric Functions

Once the concepts click, try the questions your exam asks.

Textbook exercisesEvery exercise, with model answersPast board questionsAsked from this chapter, with solutions