Trigonometry

• Core. Right angle present? Pythagoras or SOHCAHTOA. That is the whole toolkit, and bearings are just a right-angled triangle measured from north. Extended adds. No right angle? Sine rule, cosine rule or \(\tfrac12ab\sin C\). Three dimensions? Cut out a flat right-angled triangle first, then start again. And no angle question is finished until you have asked whether a second answer exists.
• Every one of these is a decision, not a calculation. Which is why the routine in the next section is written down and used out loud on every question, until it becomes invisible.
• Notice how many of these are decisions rather than calculations: which side is the hypotenuse, which angle is the reference, whether a pair is complete, which symmetry rule the curve obeys, which line is the projection. That is why the T-R-I-G routine spends three of its four steps before any formula is written.
• Right angle → Pythagoras if only lengths, a ratio if an angle is involved. Reference angle before any label. Bearing → draw the north line, then it is an ordinary right-angled triangle. Round once, at the end.
• Opposite pair → sine rule. Included angle or three sides → cosine rule. Two sides and a non-included angle → sine rule, then test the supplementary angle. An equation on \(0^\circ\) to \(360^\circ\) → inverse key, then the symmetry rule for that curve. A solid → two flat triangles, and name the projection before choosing a ratio. Special angle or no calculator → the exact table.
• On the Core route, the last hour is best spent on the two decisions that start every question: which side is the hypotenuse, and which angle is the reference. On the Extended route, add 6.5’s ambiguous case and 6.6’s projection — both are decision errors that produce confident, well-presented, wrong answers, and 6.4 is where the first of them is explained.

• I can identify the hypotenuse from the position of the right angle, not from the longest line in the drawing.
• I can find the hypotenuse from two shorter sides using \(c=\sqrt{a^2+b^2}\).
• I can rearrange to find a shorter side using \(a=\sqrt{c^2-b^2}\), and I subtract in the correct order.
• I can leave an exact answer as a surd when no rounding is requested, and give 3 significant figures when a decimal is asked for.
• I can solve a two-dimensional context problem by first establishing where a right angle actually is.
• I can check my answer: the hypotenuse must be the longest side, and a calculated shorter side must be smaller than the hypotenuse.
• I can state \(\sin\theta\), \(\cos\theta\) and \(\tan\theta\) as opposite/hypotenuse, adjacent/hypotenuse and opposite/adjacent, for an acute angle.
• I choose a reference angle before labelling any side opposite or adjacent, and I know both labels swap if the reference angle swaps.
• I can select the single ratio that contains the side I know and the side I want, then rearrange it correctly.
• I can find an angle with \(\sin^{-1}\), \(\cos^{-1}\) or \(\tan^{-1}\), and I confirm the calculator is in degree mode first.
• I can combine Pythagoras with a ratio in a multi-step two-dimensional problem.
• I can work with a three-figure bearing measured clockwise from north, resolve it into a northward and an eastward distance, and find a back bearing.
• I give an angle to 1 decimal place and any other non-exact answer to 3 significant figures, unless the question says otherwise.
• I know that the perpendicular distance from a point to a line is the shortest distance to that line, and I can calculate it.
• I can carry out calculations involving an angle of elevation, measured upward from a horizontal, and an angle of depression, measured downward from a horizontal, and transfer between two parallel horizontals using alternate angles.
• I know the exact values of \(\sin x\) and \(\cos x\) for \(x=0^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\) and \(90^\circ\), as fractions and surds.
• I know the exact values of \(\tan x\) for \(x=0^\circ\), \(30^\circ\), \(45^\circ\) and \(60^\circ\), and I can say why \(90^\circ\) is not on that list.
• I can rebuild the whole table from the half-equilateral triangle and the right isosceles triangle, rather than relying on memory alone.
• I can use an exact value to produce an exact answer such as \(6\sqrt3\), with no calculator.
• I can recognise, sketch and interpret \(y=\sin x\), \(y=\cos x\) and \(y=\tan x\) for \(0^\circ\le x\le360^\circ\), with their intercepts, maxima, minima and asymptotes in the right places.
• I can solve an equation such as \(\sin x=\tfrac{\sqrt3}{2}\) or \(2\cos x+1=0\) on \(0^\circ\le x\le360^\circ\), and give every solution in the interval.
• I can use \(\sin x=\sin(180^\circ-x)\), \(\cos x=\cos(360^\circ-x)\) and \(\tan x=\tan(x+180^\circ)\) to find the solution the calculator does not show.
• I label a triangle conventionally, so side \(a\) lies opposite angle \(A\).
• I can use the sine rule when a side is paired with the angle opposite it.
• I can use the cosine rule for two sides with their included angle, and its rearrangement \(\cos C=\dfrac{a^2+b^2-c^2}{2ab}\) when all three sides are known.
• I read a negative cosine as an obtuse angle rather than as a mistake.
• I can find an area with \(\tfrac12ab\sin C\), and I check that \(C\) is the angle enclosed between the two sides I used.
• I test the ambiguous case whenever two sides and a non-included angle are given, and I can say whether there are no, one or two triangles.
• I know that these three formulas are printed on the Extended List of formulas, and that nothing else in this topic is.
• I can reduce a solid to one or more planar right-angled triangles in stages.
• I can calculate a face diagonal, a base diagonal and a space diagonal, using Pythagoras twice where necessary.
• I can state that the angle between a line and a plane is the angle between the line and its perpendicular projection onto that plane.
• I can identify and mark that projection correctly, and I never use a vertical edge, an unrelated diagonal, or the line itself in its place.
• I can find an angle in a square-based pyramid, distinguishing a slant edge from the slant height of a face.
• I never measure a three-dimensional drawing to obtain a length or an angle.

Why this chapter repays care. Almost every mark lost in trigonometry is lost before the calculator is touched. A hypotenuse identified from the picture instead of from the right angle, an “opposite” side labelled before a reference angle was chosen, a ratio rearranged the wrong way when the unknown sits underneath — each is a decision error, and each is preventable by the routine further down this page. Extended candidates add three more of them. A sine rule started without an opposite pair, an inverse-sine answer accepted as the only possibility, and a space diagonal used as its own projection. All three are decisions too, and all three are rehearsed here.

• “The longest line in the picture is the hypotenuse.” Fix The hypotenuse is the side opposite the right angle. In a diagram that is not to scale, the longest drawn line may be anything at all. Find the right-angle marker first, then look straight across from it.
• “Opposite and adjacent are properties of the triangle.” Fix They are properties of the reference angle. Swap to the other acute angle and the two labels swap with it. Only the hypotenuse stays put.
• “Pythagoras works on any triangle if I am careful.” Fix It does not. \(a^2+b^2=c^2\) requires a right angle that is either given or properly established. On the Core route, a question with no right angle anywhere always has one waiting to be constructed — a perpendicular height, a bisected isosceles base, a rectangle’s corner.
• “I will round as I go to keep the numbers tidy.” Fix Rounding an intermediate angle or length and then feeding it into the next step drifts the final answer outside the accepted range. Store the value or leave it on the display; round once, at the end.
• “A bearing is just an angle, so two figures will do.” Fix A bearing takes three figures and is measured clockwise from north: \(048^\circ\), never \(48^\circ\). And because it is measured from north rather than from the horizontal, the northward component is the adjacent side.
• “The sine rule always starts.” E6.5 Fix It starts only when you have a side paired with the angle opposite it. Two sides and the angle between them give you no such pair, so the sine rule has nothing to stand on and the cosine rule is the only way in.
• “The calculator gave me the angle, so that is the angle.” E6.4 E6.5 Fix \(\sin^{-1}\) returns only one value, and on \(0^\circ\) to \(360^\circ\) there are usually two. Whenever you solve a sine equation, or find an angle from two sides and a non-included angle, \(180^\circ-\theta\) is a second candidate that must be tested rather than ignored or automatically accepted. Section 6.4 shows why it exists; section 6.5 shows how to test it.
• “In the cosine rule, \(C\) is just the angle I know.” E6.5 Fix \(C\) must be the angle enclosed between the two sides \(a\) and \(b\) that appear in the same formula, and \(c\) must be the side opposite it. The same requirement applies to \(\tfrac12ab\sin C\) for the area.
• “In 3D, the space diagonal is the angle’s base line.” E6.6 Fix The angle between a line and a plane is measured to the line’s perpendicular projection on that plane. For a cuboid’s space diagonal, that projection is the base diagonal — not the space diagonal, not a vertical edge, not an edge of the base.
• Assuming the longest line shown is the hypotenuse. Fix The hypotenuse is defined by position, not by length: it is the side opposite the right angle. Diagrams are not to scale, so find the right-angle marker and look straight across from it.
• Labelling opposite and adjacent before choosing a reference angle. Fix Choose the angle first. Both labels swap if you switch to the other acute angle; only the hypotenuse is fixed.
• Subtracting instead of taking the root: writing \(13-5=8\) for a shorter side. Fix \(\sqrt{13^2-5^2}=\sqrt{144}=12\). Square first, subtract second, root last.
• Multiplying when the unknown is on the bottom. Fix From \(\sin12^\circ=\dfrac{1.4}{L}\) you get \(L=\dfrac{1.4}{\sin12^\circ}=6.73\), not \(1.4\sin12^\circ=0.291\). The size check kills the wrong one instantly: a hypotenuse cannot be shorter than a side.
• Writing a bearing with two figures. Fix Bearings take three figures: \(048^\circ\), not \(48^\circ\). Add \(180^\circ\) for a back bearing under \(180^\circ\), subtract it for one over.
• Rounding an intermediate value and reusing it. Fix Keep the full display, or use the calculator’s memory. Rounding \(52.0611\ldots\) to \(52\) before the next step can move a final answer outside the accepted range.
• Measuring a diagram to obtain a length or an angle. Fix Examination diagrams are generally not to scale, and three-dimensional ones are projections in which right angles do not look like right angles. Calculate; never measure, unless the question explicitly asks for an accurate drawing.
• Working in radians without noticing. Fix Check the mode indicator, or test \(\sin30\): it must display \(0.5\). Radians are not part of this syllabus, so any radian answer is simply a machine-setting error — one that costs every angle mark in the question.
• Applying Pythagoras to a triangle with no right angle. Extended E6.5 Fix If you cannot point at the right angle, use the cosine rule. Pythagoras is the cosine rule’s special case, not a general tool.
• Using an angle of depression as the triangle’s angle at the observer. Extended E6.2 Fix It is measured from the horizontal, so at the top the triangle’s angle is \(90^\circ\) minus it. Transfer it to the far end by alternate angles, where it is the triangle’s angle. For 45 m and \(27^\circ\), the answer is 88.3 m, not 22.9 m.
• Starting the sine rule without a complete opposite pair. Extended E6.5 Fix Every fraction in the sine rule needs a side over the sine of the angle facing it. Two sides with the angle between them is a cosine-rule problem, not a sine-rule one.
• Accepting the inverse-sine value as the only possible angle. Extended E6.5 Fix Whenever the sine rule produces an angle from two sides and a non-included angle, write \(180^\circ-B\) as well and test both against a positive third angle. For \(a=8\), \(b=11\), \(A=35^\circ\), both \(52.1^\circ\) and \(127.9^\circ\) are correct answers.
• Using the wrong angle in the cosine rule or the area formula. Extended E6.5 Fix In both \(c^2=a^2+b^2-2ab\cos C\) and \(\tfrac12ab\sin C\), the angle \(C\) must be enclosed by the two sides \(a\) and \(b\) you substituted. Mark the included angle on the diagram before you write the formula.
• Treating a negative cosine as an error. Extended E6.5 Fix \(\cos C=-0.05\) is a valid result and means the angle is obtuse: \(C=92.9^\circ\). Do not take the modulus, and do not subtract from \(180^\circ\) — \(\cos^{-1}\) has already done that.
• Applying a 2D formula to three lengths that are not in one plane. Extended E6.6 Fix In a cuboid, the edges 6, 8 and 12 do not form a triangle. Do Pythagoras twice: base diagonal first (\(10\)), then the space diagonal (\(\sqrt{244}\)).
• Using the space diagonal as its own projection. Extended E6.6 Fix A line’s projection is a different line, lying in the plane. Project it first, name the projected segment, and only then choose a ratio.
• Measuring a 3D angle from the vertical instead of from the plane. Extended E6.6 Fix The two are complementary. If you have found \(39.8^\circ\) where \(50.2^\circ\) was wanted, you used the vertical edge as your second line. Subtract from \(90^\circ\), or redo it with the projection.
• Confusing a pyramid’s slant edge with its slant height. Extended E6.6 Fix The slant edge reaches a corner and projects onto half the base diagonal; the slant height reaches an edge midpoint and projects onto half the base side. For a 10 cm base and height 12 cm they are 13.9 cm and 13 cm.
• Writing \(0.5\) where an exact \(\tfrac12\) was asked for. Extended E6.3 Fix \(\sin30^\circ=\tfrac12\) exactly; \(0.5\) is a decimal that happens to be equal to it, and on a non-calculator paper you could not have obtained it. When a question says “exact”, or when you are on Paper 2, the answer keeps its fraction or its surd: \(12\sin60^\circ=6\sqrt3\), not \(10.4\).
• Giving one solution where the interval admits two. Extended E6.4 Fix On \(0^\circ\le x\le360^\circ\) a horizontal line usually cuts the curve twice. Write the calculator’s value, then its partner — \(180^\circ-x_1\) for sine, \(360^\circ-x_1\) for cosine, \(x_1+180^\circ\) for tangent — and discard the partner only if it falls outside the interval.
• Using the sine symmetry rule on a cosine or tangent equation. Extended E6.4 Fix The three rules are different because the three graphs are different. For \(\tan x=1\) the answers are \(45^\circ\) and \(225^\circ\), not \(45^\circ\) and \(135^\circ\): substituting back gives \(\tan135^\circ=-1\), which settles it in two seconds.

• One machine check before anything else. Type \(\sin 30\) into your calculator. If the display does not read \(0.5\), the calculator is not in degree mode, and every angle you produce in this chapter will be wrong while every line of your method is right. This is a setting check, not knowledge. Confirming that a machine prints \(0.5\) is not the same as knowing that \(\sin30^\circ=\tfrac12\) exactly. Knowing the exact values, and being able to use them where no calculator is allowed, is a separate Extended requirement (E6.3) taught in section 6.3. Two of the four examination papers — Paper 1 for Core and Paper 2 for Extended — do not allow a calculator at all, so on those the mode check is irrelevant and the exact values are everything.
• Reading the instruction, not guessing it. “Give your answer in its simplest form” and “leave your answer in surd form” both forbid a decimal. “Give your answer correct to 3 significant figures” forbids a surd. Neither instruction is decoration: an answer in the wrong form has not answered the question that was set, however sound the working above it.
• A one-second sanity test for any ratio. For an acute angle, \(\sin\theta\) and \(\cos\theta\) always lie between 0 and 1, because a shorter side divided by the hypotenuse cannot exceed 1. If a rearrangement leaves you asking for \(\sin^{-1}(1.16)\), the rearrangement is wrong. \(\tan\theta\) has no such limit and grows without bound as \(\theta\) approaches \(90^\circ\). Extended note (E6.5). There is one other way to meet \(\sin^{-1}\) of a number above 1, and it is not a rearrangement error: in the ambiguous case it means the triangle you were described does not exist. See the ambiguous-case clinic.
• Why sine and cosine look swapped in a bearing question. Most of the practice you have done measures an angle from a horizontal, which makes the horizontal side the adjacent one. A bearing is measured from north, so the northward side is adjacent and the eastward side is opposite. Nothing about the ratios has changed — the reference line moved, and “adjacent” moved with it. Draw the north line first and the labelling follows.
• The one reversal to remember. Angles of elevation and depression are measured from a horizontal, so the horizontal side is adjacent. Bearings, which you met in the Core half of this lab, are measured from north, so the northward side is adjacent. Sine and cosine therefore appear to swap roles between the two contexts. They have not swapped: the reference line moved, and “adjacent” moved with it.
• What earns the marks. Show \(\sin B\) as a number, write both \(B_1\) and \(180^\circ-B_1\), and show the third-angle test that keeps or kills each one. A question that says “find the obtuse angle B” has told you which to keep; one that says “find all possible values” expects both; one that supplies a diagram with a visibly acute angle at B has restricted you to the acute candidate. Read the wording before deciding what to write down.
• A quick plausibility test in a cuboid. If the height is greater than the base diagonal, the space diagonal leans steeply and its angle with the base exceeds \(45^\circ\); if the height is smaller, the angle is under \(45^\circ\). Here \(12>10\), so the answer had to be more than \(45^\circ\). \(39.8^\circ\) fails that test on sight.
• Give angles to 1 decimal place and other non-exact answers to 3 significant figures unless a question asks for an exact value. Keep every intermediate value unrounded, and write bearings with three figures.
• Give angles to 1 decimal place and other non-exact answers to 3 significant figures unless a question asks for an exact value. Questions 9 and 10 are non-calculator questions: answer them the way Paper 2 would require.
• Marking your own work honestly. Give yourself a method step only if that step is visible on your page, not merely in your head. That is the whole point of the exercise: the syllabus notes that working should be shown where it is needed to establish a method, so anything you did silently is anything you cannot be credited for. In question 5 that means writing both candidate angles down; in question 8 it means naming each projection before any calculation happens.

• Every candidate takes exactly two written papers, and which two depends entirely on the route. The papers are not interchangeable between routes, and the numbering is not intuitive: Paper 1 and Paper 2 are the two non-calculator papers, one for each route.
• Both of your papers may assess any of the Topic 6 rows your route owns. For Core that means C6.1 and C6.2 only. For Extended it means all six rows, and the four Extended-only rows carry the larger share of the marks in this topic.
• Extended E6.3 E6.4 E6.5 Paper 2 has to test Topic 6 without a calculator, so it leans on the three things that need none. Exact values (E6.3) let a ratio be evaluated on paper. Trigonometric equations (E6.4) are set at exact values, so \(\sin x=\tfrac{\sqrt3}{2}\) is a fair non-calculator question and \(\sin x=0.6\) is not. And the cosine rule becomes arithmetic when the included angle is \(60^\circ\) or \(120^\circ\), because \(\cos60^\circ=\tfrac12\) exactly. If you are on the Extended route, the exact-value table is not decoration — without it there is no way to evaluate a trigonometric ratio on Paper 2 at all.
• All angles are given in degrees and all answers are written in degrees. Confirm degree mode before you begin on a calculator paper.
• Unless the question instructs otherwise, give an angle to 1 decimal place and any other non-exact answer to 3 significant figures.
• Where the question asks for an exact answer, leave the surd or the fraction: \(\sqrt{67}\) is exact, \(8.19\) is not.