A small and completely closed Math type (3.0% of the bank) that repays study out of proportion to its share. Covers the Pythagorean theorem and the triples worth recognising on sight, labelling opposite and adjacent from the NAMED angle before using SOH-CAH-TOA, the two special right triangles, and the complementary-angle identity that sine of an angle equals cosine of its complement — with six worked examples, four traps and three checks.
Subject: SAT Prep · 61 slides · applied lesson
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Title
SAT Math · Type 15 of 19
3.0% of the question bank — 51 of 1675 questions
Objectives
Right triangles, the three trigonometric ratios, and two special triangles whose side ratios are fixed. This is the most closed topic on the Math section: the complete list of facts fits on one card, and once you have them the questions become recall rather than reasoning. That is why three per cent of the section is worth more than three per cent of your preparation time.
The single most important habit: label the sides from the angle named in the question, not from the orientation of the drawing.
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — 51 tagged questions of this type in the site's bank
Section
Section 1
Concept
Right triangles, the three trigonometric ratios, and two special triangles whose side ratios are fixed. This is the most closed topic on the Math section: the complete list of facts fits on one card, and once you have them the questions become recall rather than reasoning. That is why three per cent of the section is worth more than three per cent of your preparation time.
You will see it phrased in these ways:
The third phrasing is the one to prepare for specially. It is tested directly, it needs no triangle at all, and students who have not met it usually cannot start.
Picture it
The same three-panel card as the survey deck, so the shorthand carries over: what identifies it, what you write first, and what is built to catch you.
Figure (svg): Right triangles and trigonometry: the tell, the move, and the trap
The red panel is the whole difficulty. The hypotenuse never moves, but opposite and adjacent swap places the moment you look at the other acute angle.
Prediction
One side never changes label, and it is the anchor for everything else.
Predict first
In a right triangle, the hypotenuse is which side?
Correct: The side opposite the right angle
Why: The hypotenuse is always opposite the right angle, and it is always the longest side. Crucially, it does not depend on which acute angle the question names — unlike opposite and adjacent, which swap when you switch angles. Its position on the page is irrelevant; a rotated triangle still has its hypotenuse opposite the right angle.
Concept
Three shapes of question, and the last needs no triangle at all.
| variant | what you use | how it looks |
|---|---|---|
| Two sides, find the third | the Pythagorean theorem, or a triple | a right triangle with two lengths marked |
| An angle and a side | SOH-CAH-TOA, or a special triangle | a ladder, ramp or elevation story |
| A trig identity | sine equals cosine of the complement | given sin x, find cos of 90 minus x |
The third variant is pure recall and takes about ten seconds once you know the identity. It is the best return in the type.
Definition probe
Everything depends on doing this relative to the named angle.
Sort into buckets
In a right triangle with acute angle A, which side is which?
Discrimination
SOH-CAH-TOA, chosen by what you have and what you want.
Sort into buckets
Which ratio connects the sides in question?
Warm-up
Try it, and see whether you needed the theorem.
Discussion prompt
A right triangle has legs of 6 and 8. What is the hypotenuse?
Hint: Is this a multiple of a triple you already know?
Answer:
The hypotenuse is 10. This is a 3-4-5 triangle scaled by 2, so the sides are 6, 8 and 10.
The theorem confirms it: 36 plus 64 is 100, and the square root of 100 is 10.
Why recognising the triple matters: it turns a two-step calculation into an immediate answer, and it removes the chance of an arithmetic slip under time pressure.
The triples worth knowing: 3-4-5, 5-12-13, 8-15-17, and every multiple of each. A 9-12-15 triangle is 3-4-5 tripled.
Pattern
Three steps, and the first two happen before any trigonometry.
Check whether it is a recognisable triple or a special triangle.
Why: A 3-4-5 or a 30-60-90 turns the question into recall. This check costs two seconds and frequently ends the question.
Label the three sides relative to the angle the question NAMES.
Why: Opposite and adjacent depend entirely on which acute angle you are working from, and the drawing's orientation is irrelevant.
Pick the ratio linking the side you know to the side you want, then solve.
Why: Two sides are involved in every question — one given and one wanted — and they determine which of the three ratios applies.
Step 1 is the one students skip. On the SAT the numbers are usually chosen so a triple or a special triangle applies, and noticing saves most of the work.
Section
Section 2
Concept
In a right triangle, the squares of the two legs sum to the square of the hypotenuse.
\[ a^2 + b^2 = c^2 \]
| triple | check | common multiples |
|---|---|---|
| 3-4-5 | 9 plus 16 is 25 | 6-8-10, 9-12-15, 15-20-25 |
| 5-12-13 | 25 plus 144 is 169 | 10-24-26, 15-36-39 |
| 8-15-17 | 64 plus 225 is 289 | 16-30-34 |
| 7-24-25 | 49 plus 576 is 625 | 14-48-50 |
The SAT reuses these triples constantly. Spotting one converts a calculation into a recall and eliminates a chance to slip.
Concept
Opposite and adjacent are defined relative to the angle you are working from; the hypotenuse never changes.
Write the three labels onto the figure before choosing a ratio. Doing it in your head is where this type goes wrong.
Prediction
Recall beats calculation.
Predict first
A right triangle has a leg of 9 and a hypotenuse of 15. What is the other leg?
Correct: 12
Why: This is a 3-4-5 triangle scaled by 3, giving 9-12-15. The theorem confirms it: 225 minus 81 is 144, and the square root of 144 is 12. The answer 6 comes from subtracting 9 from 15 rather than working with squares, and the square root of 306 comes from ADDING the squares when the hypotenuse was already known.
Concept
Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent.
| ratio | definition | involves the hypotenuse? |
|---|---|---|
| sine | opposite over hypotenuse | yes |
| cosine | adjacent over hypotenuse | yes |
| tangent | opposite over adjacent | no |
That last point connects this type to similarity: all right triangles with the same acute angle are similar, which is exactly why the ratios depend only on the angle.
Prediction
Opposite and adjacent swap.
Predict first
In a right triangle, the side of length 5 is opposite angle A. Relative to the OTHER acute angle B, what is that side?
Correct: adjacent
Why: The two acute angles sit at opposite ends of each leg, so a side opposite one of them is adjacent to the other. The hypotenuse is the only side whose label does not change, because it is defined relative to the right angle rather than to either acute angle. This swap is exactly what makes labelling from the named angle so important.
Concept
A 45-45-90 has sides in the ratio 1 to 1 to root 2; a 30-60-90 has 1 to root 3 to 2.
| triangle | side ratio | which side is which |
|---|---|---|
| 45-45-90 | 1 : 1 : root 2 | the two legs are equal; the hypotenuse is root 2 times a leg |
| 30-60-90 | 1 : root 3 : 2 | 1 opposite the 30, root 3 opposite the 60, 2 is the hypotenuse |
The pairing of side with angle is what students get wrong: in a 30-60-90 it is the side opposite 30 that is shortest, not the side touching it.
Concept
For any acute angle, the sine of that angle equals the cosine of ninety minus it.
\[ \sin(x) = \cos(90^\circ - x) \]
This is tested directly and surprises people. It is worth ten seconds of recall and it is the single best-value fact in the type.
Prediction
No triangle required.
Predict first
If sin of x equals 0.8, what is cos of (90 degrees minus x)?
Correct: 0.8
Why: Sine of an angle equals cosine of its complement, so cos of (90 minus x) equals sin of x, which is 0.8. No triangle, no Pythagorean theorem, and no calculation. The answer 0.6 comes from finding cos of x itself, using a 3-4-5 triangle — a genuine quantity, and the wrong one.
Concept
Both are measured from the horizontal, and they are equal between the same two points.
The common error is measuring from the vertical. Both angles are from the horizontal, always.
Concept
Two right triangles with the same acute angle are similar, so their trigonometric ratios are identical.
So a question giving sin of x equals 3 over 5 has effectively handed you a 3-4-5 triangle, whatever the actual sizes are.
Prediction
Which side goes with which angle.
Predict first
In a 30-60-90 triangle, the hypotenuse is 10. What is the side opposite the 30-degree angle?
Correct: 5
Why: In a 30-60-90 the ratio is 1 to root 3 to 2, with the hypotenuse as the 2. So the side opposite the 30-degree angle is half the hypotenuse, which is 5. The side opposite the 60-degree angle is 5 root 3, which is the third choice — a correct length in this triangle, paired with the wrong angle.
Two truths and a lie
Three of these trigonometry statements are correct. The one left standing is false.
Eliminate the wrong options
Which statement is FALSE?
Survives elimination: c
Why: Scaling a triangle leaves every trigonometric ratio unchanged, because both the numerator and the denominator scale by the same factor. That is precisely why sine, cosine and tangent depend only on the ANGLE and not on the size of the triangle — and it is what makes trigonometry usable at all. A doubled 3-4-5 triangle is 6-8-10, and 6 over 10 equals 3 over 5.
Check
Look for a triple first.
Check your understanding
A right triangle has a hypotenuse of 26 and one leg of 10. What is the other leg?
Answer: A
Why: This is a 5-12-13 triple doubled, giving 10-24-26. The theorem confirms it: 676 minus 100 is 576, and the square root of 576 is 24.
Choice D is the most instructive error: it applies the theorem correctly in the wrong direction. When the hypotenuse is known, you subtract; when both legs are known, you add.
Section
Section 3
Worked example
A right triangle has legs of 5 and 12. Find the hypotenuse.
Figure (svg): A right triangle with legs of five and twelve and hypotenuse thirteen
Check for a triple first: 5 and 12 are the legs of the standard 5-12-13 triangle.
Why: Recognition turns the question into recall and avoids the arithmetic entirely.
So the hypotenuse is 13.
Why: The triple gives the third side directly.
Confirm with the theorem: 25 plus 144 is 169.
Why: The square root of 169 is 13, matching the recalled triple.
The check that the hypotenuse is longest catches the commonest structural error: solving for a leg when the hypotenuse was wanted, or the reverse.
Verify: check the hypotenuse is the longest side: 13 exceeds both 5 and 12.
Why: The side opposite the right angle must be longest, so the answer is structurally sound.
Answer: the hypotenuse is 13
Worked example
A ladder leans against a wall at 65 degrees to the ground. Its foot is 3 metres from the wall. How long is the ladder?
Figure (svg): A ladder against a wall forming a right triangle with the ground
Label from the 65-degree angle: the 3 m is adjacent, and the ladder is the hypotenuse.
Why: The distance along the ground touches the angle, and the ladder is opposite the right angle.
The two sides involved are adjacent and hypotenuse, so use cosine.
Why: CAH: cosine is adjacent over hypotenuse.
Write cos 65 equals 3 over L, so L equals 3 divided by cos 65, which is about 7.1 metres.
Why: Rearranging isolates the unknown hypotenuse.
Dividing rather than multiplying is the step to watch. If your hypotenuse comes out shorter than a leg, you have the ratio upside down.
Verify: check the hypotenuse exceeds the adjacent side: 7.1 is greater than 3.
Why: The hypotenuse is always longest, so a value below 3 would have signalled an inverted ratio.
Answer: about 7.1 metres
Worked example
A square has side 6. What is the length of its diagonal?
Figure (svg): A square cut along its diagonal forming a 45-45-90 triangle
The diagonal cuts the square into two 45-45-90 triangles, with the two sides as legs.
Why: A square's angles are 90, and the diagonal bisects them into two 45s.
In a 45-45-90 the ratio is 1 to 1 to root 2, so the hypotenuse is root 2 times a leg.
Why: The special triangle gives the ratio without any computation.
So the diagonal is 6 root 2, which is about 8.49.
Why: Multiplying the leg by root 2 gives the hypotenuse.
Recognising the 45-45-90 saved the squaring and the simplification. On the SAT, a square's diagonal is nearly always this pattern.
Verify: confirm with the theorem: 36 plus 36 is 72, and the square root of 72 is 6 root 2.
Why: The theorem reproduces the special-triangle result exactly.
Answer: 6 root 2, about 8.49
Step zero
Before reaching for a calculator.
Discussion prompt
A right triangle question gives you two side lengths. What do you check before applying the Pythagorean theorem?
Hint: The numbers are usually chosen deliberately.
Answer:
Check whether the numbers form a recognisable triple, or a multiple of one.
The list worth carrying: 3-4-5, 5-12-13, 8-15-17, 7-24-25, and every multiple of each.
A multiple counts: 9-12-15 is 3-4-5 tripled, and 10-24-26 is 5-12-13 doubled.
Why it matters: the SAT chooses these numbers deliberately, so recognising one turns a two-step calculation into an immediate answer and removes a chance to slip.
Then check for a special triangle — a 45-45-90 or a 30-60-90 — if any angles are given. Same reasoning: recall beats calculation.
Worked example
In a right triangle, sin of A equals 7 over 25. What is cos of B, where B is the other acute angle?
Figure (svg): A table showing the same side labelled differently from each acute angle
A and B are the two acute angles, so they are complementary: A plus B equals 90.
Why: The three angles total 180 and one of them is the right angle.
The side opposite A is adjacent to B, and the hypotenuse is shared.
Why: Switching angles swaps opposite and adjacent and leaves the hypotenuse alone.
So cos of B equals the same ratio: 7 over 25.
Why: Both are the same side over the same hypotenuse.
No lengths were needed and no triangle had to be drawn. The identity alone answers it, which is why it is worth memorising.
Verify: check with the triple: 7-24-25 is a standard triple, so the third side is 24 and cos A is 24 over 25.
Why: The two ratios differ as expected, confirming that cos B matches sin A rather than cos A.
Answer: cos of B equals 7 over 25
Worked example
From a point 40 metres from the base of a tower, the angle of elevation to the top is 32 degrees. How tall is the tower?
Figure (svg): A tower with the angle of elevation measured from a point forty metres away
Label from the 32-degree angle: the 40 m ground distance is adjacent, and the tower height is opposite.
Why: The ground distance touches the angle, and the height is across the triangle from it.
Both legs are involved and the hypotenuse is not, so use tangent.
Why: TOA: tangent is opposite over adjacent.
Write tan 32 equals h over 40, so h equals 40 times tan 32, which is about 25 metres.
Why: Multiplying isolates the opposite side.
The under-45-degrees check is genuinely useful: tangent is less than 1 below 45 degrees and greater than 1 above it, so the opposite side compares to the adjacent accordingly.
Verify: sanity-check the size: 32 degrees is well under 45, so the height should be less than the 40 m base.
Why: 25 is indeed less than 40, consistent with a tangent below 1.
Answer: about 25 metres
Faded example
From memory. Four lines that carry the type.
Fill in the blanks
Sine is opposite over hypotenuse. Cosine is adjacent over hypotenuse. Tangent is opposite over adjacent. And sine of an angle equals cosine of its complement.
Why: The fourth line is the one that is not in the mnemonic and is tested directly. It follows from the first two: the side opposite one acute angle is adjacent to the other, and both ratios share the hypotenuse, so the two expressions describe the same fraction.
Worked example
In a right triangle, tan of A equals 3 over 4. What is sin of A?
Figure (svg): A right triangle with legs three and four and hypotenuse five
Tangent is opposite over adjacent, so the opposite side is 3 and the adjacent is 4 — or any multiple.
Why: Ratios are unchanged by scaling, so any triangle with these proportions works.
Find the hypotenuse: 3 and 4 are legs of a 3-4-5 triangle, so the hypotenuse is 5.
Why: The triple gives the third side immediately.
Sine is opposite over hypotenuse, which is 3 over 5.
Why: The two sides now needed are both known.
Being given a ratio is being given a triangle. Sketching it with those two legs and finding the third side answers any other ratio for the same angle.
Verify: check cos of A is 4 over 5, and that sine squared plus cosine squared is 9 over 25 plus 16 over 25, which is 1.
Why: The identity holding confirms the triangle was constructed consistently.
Answer: sin of A equals 3 over 5
Fill the middle
Ratios and which side goes with which angle.
Fill in the blanks
A 45-45-90 triangle has sides in the ratio 1 to 1 to root 2. A 30-60-90 has sides in the ratio 1 to root 3 to 2. In a 30-60-90, the shortest side is opposite the 30 degree angle, and the hypotenuse is twice the shortest side.
Why: The third and fourth blanks are the pairings students misremember. The smallest side sits opposite the smallest angle, which is a general fact about triangles rather than something specific to the 30-60-90 — and it is the fastest way to reconstruct the pairing if it slips.
Estimation
Tangent crosses 1 at 45 degrees, which bounds a lot of answers.
Predict first
A ramp rises at 20 degrees over a horizontal run of 30 metres. Roughly how high does it rise?
Correct: about 11 metres
Why: At 45 degrees the rise would equal the run, 30 metres. Since 20 degrees is well below 45, the rise must be considerably less than 30 — which eliminates three choices immediately. The exact value is 30 times tan 20, which is about 10.9. The 45-degree benchmark is worth carrying because it bounds every tangent question without any computation.
Check
Label from the named angle, then pick.
Check your understanding
In a right triangle, angle A measures 40 degrees and the side opposite it is 12. What is the hypotenuse?
Answer: A
Why: The known side is opposite angle A and the wanted side is the hypotenuse, so the ratio linking them is sine. Writing sin 40 equals 12 over h and rearranging gives h equals 12 divided by sin 40, which is about 18.7 — sensibly larger than the 12-unit leg.
Choice B is eliminated by structure alone: any hypotenuse smaller than a known leg is wrong before you check the trigonometry.
Section
Section 4
Trap
The trap. The question names angle B, but you label opposite and adjacent from angle A because A is drawn at the bottom left.
The two acute angles sit at opposite ends of each leg, so switching between them SWAPS opposite and adjacent.
Your ratio is then the wrong one, and the answer is the reciprocal or a different side entirely.
The fix. Circle the named angle first, then write the three labels onto the figure from that angle.
The hypotenuse is the only label that does not depend on which angle you chose.
Trap
The trap. You need the hypotenuse, write cos 65 equals L over 3, and get a hypotenuse of about 1.27.
The hypotenuse is the longest side, so it cannot be shorter than the 3-metre leg.
The ratio was written upside down: cosine is adjacent over hypotenuse, so the 3 belongs on top.
The fix. Check the structural facts after computing: the hypotenuse must be longest, and a leg must be shorter than it.
If the answer violates that, the ratio was inverted.
Error analysis
A student finding a hypotenuse. The structure of the answer gives it away.
Annotate
On: \( \cos(65^\circ) = \frac{L}{3} \;\Rightarrow\; L = 3\cos(65^\circ) \approx 1.27 \)
On this type the ratio choice is usually right and its orientation is what fails. That is why the check is structural — compare the sides — rather than a re-computation.
Trap
The trap. In a 30-60-90 with hypotenuse 10, you give the side opposite the 30-degree angle as 5 root 3.
That is the side opposite the SIXTY-degree angle. The side opposite the 30 is 5, exactly half the hypotenuse.
Both lengths exist in the triangle, so the wrong one looks entirely plausible.
The fix. Use the general fact: the smallest side sits opposite the smallest angle.
So in a 30-60-90, the side opposite 30 is the shortest, and it is half the hypotenuse.
Elimination
A right triangle has a leg of 8 and a hypotenuse of 17.
Eliminate the wrong options
What is the other leg? Three choices can be ruled out without the theorem.
Survives elimination: a
Why: This is the 8-15-17 triple, so the other leg is 15. Two choices were eliminated purely on the structural fact that no leg can exceed the hypotenuse, and a third by recognising that right triangle sides do not combine additively. Checking: 64 plus 225 is 289, which is 17 squared.
Trap
The trap. Given sin x equals 0.6, a question asks for cos of (90 minus x), and you construct a triangle, find the third side, and compute cos x instead.
That gives 0.8 — a genuine ratio in the triangle, and the answer to a different question.
The identity gives 0.6 immediately, with no triangle at all.
The fix. Recognise the pattern: whenever a question pairs sine with cosine and involves 90 minus something, the identity applies.
Sine of an angle equals cosine of its complement, in both directions.
Counterexample
A rule students invent from the theorem's shape.
Discussion prompt
A student says: in any triangle, the two shorter sides squared add to the longest side squared. Give a counterexample.
Hint: Try a triangle with no right angle.
Answer:
Counterexample: a triangle with sides 4, 5 and 6. Then 16 plus 25 is 41, and 6 squared is 36. They are not equal, yet this is a perfectly valid triangle.
The theorem applies only to RIGHT triangles. The right angle is the condition, not an optional detail.
What the comparison does tell you is the type of triangle: if the sum of the squares of the two shorter sides EXCEEDS the square of the longest, the triangle is acute. Here 41 is greater than 36, so 4-5-6 is acute.
If the sum is less, the triangle is obtuse. Sides 2, 3 and 4 give 4 plus 9, which is 13, against 16 — so that triangle is obtuse.
And equality means exactly right-angled, which is the theorem itself and its converse.
Why it matters on the test: questions sometimes give three side lengths and ask whether the triangle is right-angled. That is the converse, and it is answered by testing the equality.
Edge cases
The ratios are defined for the acute angles of a right triangle. Where does that matter?
Discussion prompt
Can you use SOH-CAH-TOA on a triangle with no right angle, and what do you do when a question gives you one?
Hint: Look for a line you could draw.
Answer:
Not directly. Opposite, adjacent and hypotenuse are only defined relative to a right angle, so a triangle without one has no hypotenuse at all.
What the SAT does instead: it gives you a figure where a right triangle can be found or created — often by dropping a perpendicular from a vertex to the opposite side.
That perpendicular splits the triangle into two right triangles, and the ratios apply within each.
A second common construction: a rectangle or square cut along a diagonal, which always produces right triangles.
The practical rule: if a question involves trigonometry and you cannot see a right angle, look for one you can draw. The SAT does not test the sine or cosine rules for general triangles.
And the reverse edge: the ratios also apply to angles of 0 and 90 in more advanced work, but on the SAT you are only ever working with the acute angles of an actual right triangle.
Check
No triangle needed.
Check your understanding
In a right triangle, angles P and Q are the two acute angles. If cos of P equals 5 over 13, what is sin of Q?
Answer: A
Why: P and Q are complementary, since the two acute angles of a right triangle sum to 90. Cosine of an angle equals sine of its complement, so sin of Q equals cos of P, which is 5 over 13. No lengths and no triangle are required.
Choice B is the trap for students who solve it by construction rather than by identity — they build the triangle correctly and then read the wrong ratio off it.
Section
Section 5
Matching
Six situations, six tools. No arithmetic.
Match the pairs
Why: Only two of these six require any trigonometry at all. Two are special triangles, one is the theorem, and one is an identity — which is a fair reflection of the type: most questions are answered by recognition rather than by computation.
Sorting
Each is a set of three side lengths.
Sort into buckets
Do they form a right triangle?
The comparison also classifies: sum of squares greater than the largest square means acute, less means obtuse, equal means right-angled. That converse is occasionally asked directly.
Comparison
Fill the blanks from memory.
Comparison matrix
| 45-45-90 | 30-60-90 | |
|---|---|---|
| Side ratio | 1 to 1 to root 2 | 1 to root 3 to 2 |
| Where it comes from | a square cut along its diagonal | an equilateral triangle cut in half |
| The hypotenuse is | root 2 times a leg | twice the shortest side |
| The shortest side is opposite | either 45 — the legs are equal | the 30 degree angle |
The bottom row is the pairing most often misremembered, and the general fact behind it is worth carrying instead: in any triangle, the smallest side sits opposite the smallest angle.
Trade off
Fill in what each approach costs.
Comparison matrix
| approach | time | risk |
|---|---|---|
| Recognise a triple | about two seconds | none, if you check it satisfies the theorem |
| Apply the Pythagorean theorem | about thirty seconds | adding when you should subtract |
| Recognise a special triangle | about two seconds | pairing the wrong side with the angle |
| Use the complementary identity | about five seconds | none — it needs no triangle at all |
Three of these four are recall rather than computation, which is the character of the whole type. The study time goes into knowing the facts, not into practising the arithmetic.
Real world
One minute on why right-triangle trigonometry is everywhere.
Discussion prompt
Surveying, navigation and construction all rest on right-triangle trigonometry. What problem does it actually solve?
Answer:
It measures distances you cannot reach. Stand a known distance from a tower, measure the angle of elevation, and the height follows from one tangent — no ladder required.
That is exactly how surveying works. A theodolite measures angles precisely, and known baselines plus angles give every other distance by trigonometry.
Navigation uses the same idea: a bearing and a distance decompose into north and east components using sine and cosine, which is how dead reckoning positions a ship.
Construction depends on the 3-4-5 triple to check that a corner is square, because a triangle with those side lengths is guaranteed right-angled by the converse of the theorem.
And the ratios work at all because of similarity: every right triangle with the same acute angle has the same ratios, so measuring a small triangle tells you about an enormous one.
Ranking
For an acute angle of 30 degrees, order these four values from smallest to largest.
Put in order
Why: From the 30-60-90 ratios: sin 30 is one half, or 0.5. tan 30 is 1 over root 3, about 0.577. cos 30 is root 3 over 2, about 0.866. tan 60 is root 3, about 1.732. Ordered: 0.5, 0.577, 0.866, 1.732. Note that tan 60 exceeds 1, which is the 45-degree benchmark in action — tangent passes 1 at 45 degrees and grows without bound after it.
Concept
This type is 3.0 per cent of the section and is completely closed, which makes it one of the highest returns per hour on the whole test.
| session | what you do | why |
|---|---|---|
| 1 | Memorise the four triples and their multiples, then find third sides for twenty triangles. | Recognition converts calculation into recall and removes arithmetic risk. |
| 2 | Fifteen SOH-CAH-TOA questions, writing OPP, ADJ and HYP onto every figure first. | Labelling from the named angle is the single most common failure. |
| 3 | Ten special-triangle questions, stating which side goes with which angle before computing. | The pairing is what students misremember, not the ratio. |
| 4 | Ten complementary-identity questions, answered without drawing anything. | Pure recall, ten seconds each, and almost nobody prepares for it. |
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — filter the bank to this skill tag — 51 questions, roughly a third of each difficulty
Explain it to yourself
Close the deck.
Discussion prompt
Without looking, state the complementary-angle identity and explain WHY it holds, using the labelling of a right triangle.
Hint: Think about the same side seen from two different angles.
Answer:
The identity: sine of an angle equals cosine of its complement, and equally, cosine of an angle equals sine of its complement.
Why: the two acute angles of a right triangle sum to 90, so each is the complement of the other.
The side opposite one acute angle is the side adjacent to the other, and both ratios are taken over the same hypotenuse.
So sin A, which is opposite over hypotenuse, and cos B, which is adjacent over hypotenuse, are the same fraction — the same side over the same hypotenuse.
If you gave the reason as well as the statement, you can reconstruct it under pressure rather than hoping to recall it.
Explain it
Two minutes, out loud.
Discussion prompt
A friend gets the right ratio and the wrong answer on trigonometry questions. What is happening, and what do you tell them?
Answer:
Diagnose it: they are labelling opposite and adjacent from the wrong angle, usually the one drawn at the bottom left rather than the one the question names.
Show why it matters. Draw a right triangle and label the sides from angle A. Then label them again from angle B. Opposite and adjacent have swapped; only the hypotenuse stayed.
Give the reason: the two acute angles sit at opposite ends of each leg, so a side across from one is touching the other.
Then give the physical habit: circle the named angle, and write OPP, ADJ and HYP directly onto the three sides before choosing a ratio.
And the closing check: the hypotenuse must be the longest side. If their answer makes a leg longer than the hypotenuse, something was labelled or inverted wrongly.
Commit first
Commit before you check.
Predict first
If cos of 25 degrees equals about 0.906, what is sin of 65 degrees?
Correct: about 0.906
Why: 25 and 65 are complementary, since they sum to 90. Cosine of an angle equals sine of its complement, so sin 65 equals cos 25, which is about 0.906. The answer 0.423 is sin 25, a genuine value and the wrong one. No triangle, no theorem and no calculator work is needed — recognising the complement is the entire question.
Connect it up
Blank paper.
Draw it
Draw a right triangle and mark one acute angle A. Label the three sides OPP, ADJ and HYP relative to A, then draw a second copy labelled relative to the other acute angle B, so the swap is visible side by side. Beside them write SOH-CAH-TOA with each ratio spelled out. Underneath, draw the two special triangles with their side ratios and mark which side sits opposite which angle. In a box, list the four triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25, and note that every multiple counts. At the bottom, write the identity: sine of an angle equals cosine of its complement, with a one-line reason — the side opposite one acute angle is adjacent to the other.
Exit ticket
One question before you close the deck.
Predict first
A question gives a right triangle with a hypotenuse of 25 and a leg of 7, and asks for the third side. What do you do first?
Correct: Check whether 7 and 25 belong to a known triple
Why: 7 and 25 are the leg and hypotenuse of the 7-24-25 triple, so the third side is 24 with no computation at all. The theorem would reach the same answer in about thirty seconds and with a chance of an arithmetic slip. A trigonometric ratio is not available since no angle is given, and special triangles apply when angles are stated rather than when three sides are in play.
Recap
One type, one card of facts: learn them and the questions become recall.
| never do this | do this instead |
|---|---|
| Label from the angle drawn at the bottom left | Circle the named angle and label from it |
| Accept a hypotenuse shorter than a leg | Check the structure: the hypotenuse is longest |
| Give the side opposite 60 when asked for the side opposite 30 | Smallest side goes with smallest angle |
| Build a triangle to find cos of 90 minus x | Apply the identity and stop |
| Add the squares when the hypotenuse is already known | Subtract: leg squared equals hypotenuse squared minus leg squared |
| Use SOH-CAH-TOA on a triangle with no right angle | Draw a perpendicular to create one |
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