Trigonometry

• Right angle present? Pythagoras or SOHCAHTOA. No right angle? Sine rule, cosine rule or \(\tfrac12ab\sin C\). Three dimensions? Cut out a flat right-angled triangle first, then start again.
• Every one of these eight 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 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 or SOHCAHTOA. Opposite pair → sine rule. Included angle or three sides → cosine rule. Two sides and a non-included angle → sine rule, then test the supplement. A solid → two flat triangles, and name the projection before choosing a ratio.
• The two subtopics that reward the last hour most are 6.3’s ambiguous case and 6.4’s projection, because both are decision errors that produce confident, well-presented, wrong answers.

• 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.
• 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 find the shortest distance from a point to a line by constructing the perpendicular, and explain why any other route is longer.
• I can use 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 can work with a three-figure bearing measured clockwise from north, and find a back bearing.
• 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 keep intermediate values unrounded and round only at the end.
• 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 sine rule started without an opposite pair, an inverse-sine answer accepted as the only possibility, a space diagonal used as its own projection — each is a decision error, and each is preventable by the routine on this page.

Right-Angled Triangles

Triangles containing an angle of ninety degrees, in which each acute angle is linked to two of the three sides by the ratios sine, cosine and tangent: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse and tangent is opposite over adjacent. The labels opposite and adjacent are defined relative to a chosen reference angle and swap when the other acute angle is chosen, while the hypotenuse, opposite the right angle, never changes. These ratios, with Pythagoras' theorem, solve two-dimensional problems including shortest distances, angles of elevation and depression, and bearings.

Pythagoras and Trigonometry in 3D

The technique of solving a three-dimensional problem by reducing it to a sequence of flat right-angled triangles, each lying wholly in one plane. A length such as the space diagonal of a cuboid is found in stages: Pythagoras on the base gives the base diagonal, and Pythagoras on the vertical triangle formed by that diagonal and the height gives the space diagonal. The angle between a line and a plane is defined as the angle between the line and its perpendicular projection onto that plane, so the projection must be identified and marked before any ratio is used. No two-dimensional formula may be applied to three lengths that do not lie in a single plane.

Pythagoras’ Theorem

The relationship between the three sides of a right-angled triangle: the square on the hypotenuse, the side opposite the right angle, equals the sum of the squares on the two shorter sides. Written a squared plus b squared equals c squared, with c the hypotenuse, it converts any two known sides into the third, and it applies only where a right angle is given or has been established.

Non-Right-Angled Triangles

Triangles containing no right angle, solved with three formulas supplied on the examination paper. The sine rule, a over sine A equals b over sine B, applies when a side is known together with the angle opposite it. The cosine rule, c squared equals a squared plus b squared minus 2ab cos C, applies to two sides with their included angle, and rearranges to give an angle when all three sides are known. The area is half ab sin C, where C is the angle enclosed between sides a and b. Conventional labelling places side a opposite angle A, and where two sides and a non-included angle are given the sine rule produces two candidate angles that must both be tested.

• “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. With no right angle the cosine rule is the correct tool — and it reduces to Pythagoras when \(C=90^\circ\), because \(\cos 90^\circ=0\).
• “The sine rule always starts.” 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.” Fix \(\sin^{-1}\) returns only the acute value. Whenever you were given two sides and a non-included angle, \(180^\circ-B\) is a second candidate that must be tested rather than ignored or automatically accepted.
• “In the cosine rule, \(C\) is just the angle I know.” 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.
• “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.
• “In 3D, the space diagonal is the angle’s base line.” 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.
• Applying Pythagoras to a triangle with no right angle. 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.
• 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.
• Using an angle of depression as the triangle’s angle at the observer. 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.
• 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.
• Starting the sine rule without a complete opposite pair. 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. 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. 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. 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.
• 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.
• Applying a 2D formula to three lengths that are not in one plane. 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. 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. 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. 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.
• 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.

• 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.
• 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.
• Sanity check on the cosine rule. Put \(C=90^\circ\) into \(c^2=a^2+b^2-2ab\cos C\). Since \(\cos90^\circ=0\), the last term vanishes and you are left with \(c^2=a^2+b^2\) — Pythagoras. That is not a coincidence; Pythagoras is the right-angled special case of the cosine rule. If you ever mis-copy the formula, this substitution will tell you.
• 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 — or, in 6.3, the triangle does not exist. \(\tan\theta\) has no such limit and grows without bound as \(\theta\) approaches \(90^\circ\).
• The one reversal to remember. Angles of elevation and depression are measured from a horizontal, so the horizontal side is adjacent. Bearings 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.
• 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.

• Both papers may assess any of 6.1 to 6.4. The difference is not which subtopics appear but what kind of numbers appear in them, and what you are expected to do with your answer at the end.
• All angles are in degrees. Confirm degree mode before you begin.
• Unless the question instructs otherwise, give an angle in Paper 2 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: \(\sqrt{67}\) is exact, \(8.19\) is not.
• A bearing is written with three figures: \(062^\circ\), not \(62^\circ\).
• Intermediate values are never rounded. Rounding \(52.061\ldots^\circ\) to \(52^\circ\) before using it again drifts the final answer, and the result is close enough to look right while falling outside the accepted range.