Trigonometric ratios in right-angled triangles
Using sine, cosine and tangent to find a missing side or a missing angle in a right-angled triangle
About three lessons of 45 to 60 minutes
Measuring what you cannot reach
You cannot climb a tall tree with a tape measure, but you CAN measure how far you are standing from its base, and the angle from the ground up to its top. That is enough. Trigonometric ratios connect an ANGLE in a right-angled triangle to a ratio of 2 of its sides, so knowing 1 side and 1 angle unlocks every other side, and knowing any 2 sides unlocks the angle.
3 ratios do all the work: sine, cosine and tangent, remembered with SOH-CAH-TOA. Crucially, each ratio depends ONLY on the angle, never on the triangle's size, because any 2 right-angled triangles with the same angles are similar (same shape, proportional sides). A tiny right triangle and a huge one with the same angle always give exactly the same sin, cos and tan.
- The height of a tree, from its shadow's angletrigonometry finds height without climbing
- How far a ladder reaches up a wallhypotenuse and angle known, opposite side (height) unknown
- A lighthouse's angle of sight to a distant ship2 sides known, the angle itself unknown
- A ramp's angle for a given rise and runthe same tangent ratio builders use for accessibility ramps
What students will be able to do
Students will label the opposite, adjacent and hypotenuse sides of a right-angled triangle relative to a given angle, calculate sin, cos and tan as ratios of sides, use a trigonometric ratio to find a missing side given an angle, and use an inverse trigonometric ratio to find a missing angle given 2 sides.
- I can identify the opposite, adjacent and hypotenuse sides of a right-angled triangle, relative to a marked angle.
- I can calculate sin(theta), cos(theta) and tan(theta) from a right triangle's side lengths.
- I can explain why the trig ratios stay the same for any similar right-angled triangle.
- I can use a trigonometric ratio to find a missing side, given 1 side and 1 angle.
- I can use an inverse trigonometric ratio (asin, acos, atan) to find a missing angle, given 2 sides.
Standards this unit teaches
- KS3 Maths: Geometry and measuresUK National Curriculum (England)Geometry and measures
Statutory requirement (Department for Education, "National curriculum in England: mathematics programmes of study", updated 28 September 2021, Key stage 3, "Geometry and measures" strand, https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study): pupils should be taught to "use Pythagoras' Theorem and trigonometric ratios in similar triangles to solve problems involving right-angled triangles". The uk-year-9-pythagoras-theorem unit already builds the Pythagoras' Theorem half of this bullet, and explicitly deferred trigonometric ratios as "a distinct, larger topic for a future unit"; this is that unit.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Hypotenuse
- the longest side of a right-angled triangle, always opposite the right angle
- Opposite side
- the side directly across from the angle being considered, not touching it
- Adjacent side
- the shorter side that touches the angle being considered, but is not the hypotenuse
- Sine, cosine, tangent
- the 3 trigonometric ratios: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent
- SOH-CAH-TOA
- a memory phrase for the 3 ratios: Sin=Opp/Hyp, Cos=Adj/Hyp, Tan=Opp/Adj
- Inverse trig ratio
- asin, acos or atan: recovers the ANGLE from a known ratio of sides, the reverse of sin/cos/tan
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Labelling right triangles and finding trig ratios
ConcreteFor an angle theta in a right-angled triangle (not the right angle itself), the 3 sides get special names relative to theta: the hypotenuse (always the longest side, opposite the right angle), the opposite side (directly across from theta), and the adjacent side (next to theta, but not the hypotenuse). SOH-CAH-TOA gives the 3 ratios: sin(theta) = opposite/hypotenuse, cos(theta) = adjacent/hypotenuse, tan(theta) = opposite/adjacent.
These ratios depend ONLY on the angle, never on the triangle's size. A 3-4-5 triangle and a 6-8-10 triangle (exactly double) have identical angles, because 1 is simply a scaled-up copy of the other (they are similar triangles), so their trig ratios match exactly: 3/5 = 6/10 = 0.6.
A right-angled triangle has legs 3 and 4 and hypotenuse 5. The angle theta is opposite the side of length 3. Find sin(theta), cos(theta) and tan(theta).
- sin(theta) = opposite/hypotenuse = 3/5 = 0.6.
- cos(theta) = adjacent/hypotenuse = 4/5 = 0.8.
- tan(theta) = opposite/adjacent = 3/4 = 0.75.
Answer: sin(theta) = 0.6, cos(theta) = 0.8, tan(theta) = 0.75.
- Using the same 3-4-5 triangle, if theta were instead opposite the side of length 4, what would cos(theta) be?
- Which ratio, sin, cos or tan, never involves the hypotenuse?
2. Finding a missing side using a trig ratio
PictorialGiven 1 side and 1 angle (not the right angle), every other side can be found: opposite = hypotenuse x sin(theta), adjacent = hypotenuse x cos(theta), or opposite = adjacent x tan(theta). Choose the ratio that connects the side you know to the side you want.
Unlike Pythagoras' theorem's whole-number triples, most angles do not give whole-number sides, so a trig answer is usually a genuine calculator-based decimal approximation, correctly rounded, not an exact value.
A ladder 10 m long leans against a wall, making an angle of 30 degrees with the ground. How high up the wall does it reach, and how far is its base from the wall? Give each answer correct to 1 decimal place.
- Height (opposite the 30 degree angle) = hypotenuse x sin(theta) = 10 x sin(30) = 10 x 0.5 = 5.0 m.
- Ground distance (adjacent to the 30 degree angle) = hypotenuse x cos(theta) = 10 x cos(30) = 10 x 0.8660... = 8.7 m (1 dp).
Answer: Height = 5.0 m. Ground distance ~= 8.7 m.
- A ladder 8 m long makes a 45 degree angle with the ground. How high up the wall does it reach, correct to 1 decimal place?
- Why do most trigonometry answers need rounding, unlike Pythagoras' theorem answers built from whole-number triples?
3. Finding a missing angle using inverse trig ratios
AbstractIf 2 sides are known instead of an angle, the inverse ratios recover theta: theta = asin(opposite/hypotenuse), theta = acos(adjacent/hypotenuse), or theta = atan(opposite/adjacent). Choose the inverse ratio that matches the 2 sides you know.
The special case where both legs are equal always gives a 45 degree angle, since tan(theta) = opposite/adjacent = 1 whenever the 2 legs match, and atan(1) = 45 degrees exactly.
A ramp rises 2 m over a horizontal distance of 9 m. Find the angle the ramp makes with the ground, correct to 1 decimal place.
- The 2 m rise is opposite theta, and the 9 m horizontal distance is adjacent to theta.
- theta = atan(opposite/adjacent) = atan(2/9) = atan(0.2222).
- theta = 12.5 degrees (1 dp).
Answer: theta ~= 12.5 degrees.
- A right-angled triangle has a hypotenuse of 13 and the side opposite theta is 5. Find theta, correct to 1 decimal place.
- Which inverse ratio would you use if you know the 2 legs but not the hypotenuse?
Common misconceptions and how to address them
Misconceptionsin, cos and tan can be used on any triangle, not just right-angled ones.
Why it happens: Students see SOH-CAH-TOA as a general triangle tool rather than one that depends on a right angle existing in the first place.
How to address it: Opposite, adjacent and hypotenuse only make sense relative to a RIGHT angle: the hypotenuse is defined as the side opposite the right angle. Without a right angle, these 3 ratios do not apply (a different tool, the sine and cosine rules, is used for non-right triangles at a later stage).
MisconceptionThe opposite and adjacent sides are fixed properties of a triangle, the same whichever angle you pick.
Why it happens: Students memorise 'opposite' and 'adjacent' as if they were permanently attached to specific sides, rather than defined relative to the chosen angle.
How to address it: Opposite and adjacent are defined RELATIVE TO theta. Picking the OTHER acute angle in the same triangle swaps which leg is opposite and which is adjacent, while the hypotenuse never changes (it is always opposite the right angle).
MisconceptionTo find a missing angle, just divide the 2 given sides in whichever order looks right, then take sin of the result.
Why it happens: Students skip identifying which 2 sides they actually have (opposite/hypotenuse, adjacent/hypotenuse, or opposite/adjacent) and default to a single memorised button sequence.
How to address it: First name the 2 known sides using opposite/adjacent/hypotenuse. That naming tells you exactly which inverse ratio applies: opposite and hypotenuse means asin; adjacent and hypotenuse means acos; opposite and adjacent means atan.
MisconceptionA rounded decimal trig answer, like 12.5 degrees, means a mistake was made somewhere.
Why it happens: Students expect maths answers to be exact whole numbers, as most earlier topics have trained them to expect.
How to address it: Only a few special angles (0, 30, 45, 60, 90 degrees) give clean ratios. Every other angle gives a genuine, correctly rounded decimal from a real calculation, exactly as intended, not a sign of an error.
Guided practice (with answers)
1. A right-angled triangle has legs 5 and 12 and hypotenuse 13. Theta is opposite the side of length 5. Find sin(theta).
Answer: 5/13 ~= 0.385 (3 dp).
2. Using the same triangle (legs 5 and 12, hypotenuse 13, theta opposite the side of length 5), find tan(theta).
Answer: 5/12 ~= 0.417 (3 dp).
3. A right-angled triangle has a hypotenuse of 20 and an angle of 60 degrees. Find the side opposite the angle, correct to 1 decimal place.
Answer: 17.3, because 20 x sin(60) = 20 x 0.8660... = 17.3.
4. Using the same triangle (hypotenuse 20, angle 60 degrees), find the side adjacent to the angle.
Answer: 10, because 20 x cos(60) = 20 x 0.5 = 10 exactly.
5. A right-angled triangle has a hypotenuse of 10 and the side adjacent to theta is 6. Find theta, correct to 1 decimal place.
Answer: 53.1 degrees, because acos(6/10) = acos(0.6) = 53.1 degrees.
6. A right-angled triangle has a side of 7 opposite theta and a side of 7 adjacent to theta. Find theta.
Answer: 45 degrees exactly, because atan(7/7) = atan(1) = 45 degrees.
Independent practice worksheets
Practise trigonometric ratios with computed, never-wrong answer keys, using both exact Pythagorean-triple ratios and genuine calculator-based trig calculations.
Differentiation
- Write 'SOH-CAH-TOA' at the top of every page and label hypotenuse/opposite/adjacent with coloured pens before choosing a ratio.
- Start with only the 30 and 45 degree angles, where the arithmetic stays clean, before moving to arbitrary decimal angles.
- Keep a completed 3-4-5 triangle example visible as a reference while labelling new triangles.
- For finding a missing angle, write the ratio as a fraction first (e.g. 5/13), then convert to a decimal, then apply the inverse function, 1 step at a time.
- Investigate why sin(theta) = cos(90 - theta) for the 2 acute angles of the same right-angled triangle.
- Solve a 2-step problem needing Pythagoras' theorem AND a trig ratio within the same triangle.
- Explore angle-of-elevation and angle-of-depression problems, including cases where 2 right triangles share a side.
- Derive the exact trig ratios for the 30-60-90 and 45-45-90 triangles from first principles, without a calculator.
Assessment: exit ticket
A three-question exit ticket sampling a trig ratio, finding a missing side, and finding a missing angle.
1. A right-angled triangle has legs 6 and 8 and hypotenuse 10. Theta is opposite the side of length 6. Find tan(theta).
Answer: 0.75, because tan(theta) = opposite/adjacent = 6/8 = 0.75.
2. A right-angled triangle has a hypotenuse of 12 and an angle of 40 degrees. Find the side opposite the angle, correct to 1 decimal place.
Answer: 7.7, because 12 x sin(40) = 12 x 0.6428... = 7.7 (1 dp).
3. A right-angled triangle has a hypotenuse of 15 and the side adjacent to theta is 9. Find theta, correct to 1 decimal place.
Answer: 53.1 degrees, because acos(9/15) = acos(0.6) = 53.1 degrees.
Teacher notes and timings
- Rough timing across 3 lessons: Lesson 1 labelling triangles and finding ratios (section 1), Lesson 2 finding a missing side (section 2), Lesson 3 finding a missing angle plus the exit ticket (section 3).
- This unit is the direct sequel to uk-year-9-pythagoras-theorem, and completes the curriculum bullet that unit explicitly deferred ('trigonometric ratios are a distinct, larger topic for a future unit'). Teach them back to back if timetabling allows.
- The rightTriangle figure in every section reuses the same engine as the Pythagoras unit, with custom side labels ('opposite', 'adjacent', 'hypotenuse') instead of the generic a/b/c, so students see the labelling is relative to theta, not a fixed a/b/c naming.
- Language to keep repeating: SOH-CAH-TOA; opposite/adjacent are relative to theta, not fixed to 1 side; a rounded calculator answer is expected and correct for most angles, not a mistake.
- Present and print both work: work through the ladder and ramp problems live on the board, asking students to name opposite/adjacent/hypotenuse before choosing a ratio, then print the 3 linked worksheets for independent practice.