Right-Angled Trigonometry in 2D

SPEC G20 — GEOMETRY & MEASURES

SOH CAH TOA — using sine, cosine and tangent to find sides and angles in right-angled triangles, plus angles of elevation and depression.

What You'll Learn

Trigonometry relates the angles of a right-angled triangle to its side lengths through three ratios. SOH CAH TOA tells you which ratio to pick — once you can identify the opposite, adjacent and hypotenuse relative to a chosen angle, the rest is substituting and solving.

By the end of this lesson, you will be able to:

State the three trig ratios: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj

Identify the opposite, adjacent and hypotenuse relative to a chosen angle

Pick the correct ratio (SOH, CAH, or TOA) based on the two sides involved

Find a missing side from one side and one angle

Find a missing angle from two sides using inverse functions (sin⁻¹, cos⁻¹, tan⁻¹)

Distinguish angles of elevation (looking up) from angles of depression (looking down)

Apply trig to real-world contexts: ladders, ramps, towers, aeroplanes

Set your calculator to DEGREES mode (not radians) before any trig calculation

Why It's High Priority

Trig appears on every paper, often as a 3-mark question on Paper 2 or 3 (calculator papers) and as exact-value questions on Paper 1 (covered in Lesson 4.18). Mark schemes typically award one mark for the correct ratio, one for the substitution, and one for the answer — so even partial credit is recoverable if you set up correctly.

The Three Trig Ratios — SOH CAH TOA

For any acute angle θ in a right-angled triangle, three ratios are defined relative to that angle:

sin θ

opposite

──────

hypotenuse

SOH

cos θ

adjacent

──────

hypotenuse

CAH

tan θ

opposite

──────

adjacent

Right-Angled Only

SOH CAH TOA works only for right-angled triangles. For non-right triangles you need the sine rule or cosine rule (Higher tier — Lesson 4.18).

Identifying Sides Relative to θ

Before you can pick a ratio, you must label the three sides relative to your chosen angle θ. The labels move depending on which non-right angle you choose.

Three rules — fixed order

Hypotenuse — always opposite the right angle, always the longest side. Independent of θ.

Opposite — the leg across from the chosen angle θ.

Adjacent — the leg next to θ that is not the hypotenuse.

The Hypotenuse Doesn't Move

Whichever angle you choose, the hypotenuse stays the same — it's always the side opposite the right angle. Only the labels "opposite" and "adjacent" swap depending on which non-right angle you pick.

Choosing the Right Ratio

Look at which two sides are involved (one given, one unknown) and pick the matching ratio.

Sides involvedRatio to useAcronym

opposite & hypotenusesin θ = opp / hypSOH

adjacent & hypotenusecos θ = adj / hypCAH

opposite & adjacenttan θ = opp / adjTOA

Decision routine

Identify the chosen angle θ (the marked one, or the one in the question)

Label the three sides: hypotenuse first, then opposite (across from θ), then adjacent (the remaining leg)

Identify which two sides are involved — one given, one unknown

Pick the ratio that uses those two sides: SOH, CAH, or TOA

Substitute and solve

If the Hypotenuse Isn't Involved

If both the given side and the unknown side are legs (and the hypotenuse isn't mentioned), use tan — it's the only ratio that doesn't involve the hypotenuse.

Finding a Missing Side

When you have one angle and one side and need a different side, set up the appropriate ratio and rearrange to make the unknown the subject.

Two rearrangements

If the unknown is in the numerator of the ratio (e.g. sin θ = opp/hyp, finding opp), multiply both sides by the denominator:

opp = hyp × sin θ

Example: hyp = 10 cm, θ = 35° → opp = 10 × sin 35° = 5.74 cm.

If the unknown is in the denominator (finding hyp from opp and θ), multiply both sides by hyp then divide both sides by sin θ:

hyp = opp / sin θ

Example: opp = 6 cm, θ = 40° → hyp = 6 / sin 40° = 9.34 cm (3 s.f.). The same logic applies for cos and tan when the unknown is in the denominator.

Calculator Mode — DEGREES

Before any trig calculation, check your calculator screen displays DEG (or D). Radians mode (RAD or R) gives wildly wrong answers. To switch: most calculators use SHIFT + MODE, then select 1 (Deg). On Casio Classwiz: SHIFT + MENU → Angle Unit → Degree.

Round at the End

Keep the full calculator value through the working — don't round intermediate steps. Only round the final answer to the precision the question asks for (typically 3 s.f. or 1 d.p. for angles).

01Worked Example — Sin: Find Opposite Side

In a right-angled triangle, the hypotenuse is 10 cm and one of the angles is 35°. Find the length of the side opposite the 35° angle. Give your answer to 3 s.f. (3 marks)

MMethod. Relative to the 35° angle: hypotenuse = 10, opposite = x (unknown). The two sides involved are opposite and hypotenuse → use sin (SOH).

sin 35° = opp / hyp = x / 10

CCalculate.

Rearrange: x = 10 × sin 35°

= 10 × 0.5736… (calculator value, DEG mode)

= 5.736… cm

= 5.74 cm (3 s.f.)

EEvaluate. Sense-check: the opposite side must be less than the hypotenuse (since it's a leg, not the hypotenuse) — and 5.74 < 10 ✓. Also: 35° < 45°, so the opposite is shorter than the adjacent — visually consistent with the diagram.

x = 5.74 cm (3 s.f.)

02Worked Example — Cos: Find Adjacent Side

In a right-angled triangle, the hypotenuse is 12 cm and one angle is 25°. Find the length of the side adjacent to the 25° angle. Give your answer to 3 s.f. (3 marks)

MMethod. Relative to 25°: hypotenuse = 12, adjacent = x. The two sides involved are adjacent and hypotenuse → use cos (CAH).

cos 25° = adj / hyp = x / 12

CCalculate.

Rearrange: x = 12 × cos 25°

= 12 × 0.9063… (DEG mode)

= 10.876… cm

= 10.9 cm (3 s.f.)

EEvaluate. 25° is small, so the adjacent should be close to the full hypotenuse — and 10.9 is close to 12 ✓. Linear units (cm).

x = 10.9 cm (3 s.f.)

03Worked Example — Tan: Find Side from Two Legs

A right-angled triangle has one angle of 35°. The side adjacent to this angle is 8 cm. Find the length of the side opposite the 35° angle. Give your answer to 3 s.f. (3 marks)

MMethod. Relative to 35°: adjacent = 8 cm, opposite = x. The hypotenuse is not involved — both involved sides are legs → use tan (TOA).

tan 35° = opp / adj = x / 8

CCalculate.

Rearrange: x = 8 × tan 35°

= 8 × 0.7002… (DEG mode)

= 5.601… cm

= 5.60 cm (3 s.f.)

EEvaluate. 35° < 45°, so tan 35° < 1, meaning opposite < adjacent. And 5.60 < 8 ✓. The result should be plausible compared to a 45° triangle (where opp = adj).

x = 5.60 cm (3 s.f.)

Finding a Missing Angle

When you know two sides and want the angle, set up the same SOH/CAH/TOA ratio — but apply the inverse trig function at the end.

If you know opp & hyp

θ = sin⁻¹(opp/hyp)

If you know adj & hyp

θ = cos⁻¹(adj/hyp)

If you know opp & adj

θ = tan⁻¹(opp/adj)

The inverse keys

On your calculator, sin⁻¹, cos⁻¹, and tan⁻¹ are usually accessed via SHIFT + sin (or cos / tan). They reverse the trig function: if sin θ = 0.5, then θ = sin⁻¹(0.5) = 30°.

sin⁻¹ vs (sin)⁻¹

The notation sin⁻¹(x) means "the angle whose sine is x" — not 1/sin(x). Even though it looks like a reciprocal, it is the inverse function. The reciprocal 1/sin(x) is called cosec(x) and isn't on the GCSE spec.

04Worked Example — Find an Angle

A right-angled triangle has one leg of 5 cm and another leg of 12 cm. Find the angle θ between the 12 cm leg and the hypotenuse. Give your answer to 1 d.p. (3 marks)

MMethod. The unknown angle θ is between the 12 cm leg and the hypotenuse. Relative to θ: the 12 cm leg is the adjacent, the 5 cm leg is the opposite. Two legs → use tan (TOA).

tan θ = opp / adj = 5 / 12

CCalculate.

θ = tan⁻¹(5/12)

= tan⁻¹(0.4167…) (DEG mode)

= 22.619…°

= 22.6° (1 d.p.)

EEvaluate. The two legs 5 and 12 give a hypotenuse of 13 (the 5-12-13 Pythagorean triple from Lesson 4.15). Cross-check using sin: sin 22.6° ≈ 0.385, and 5/13 = 0.385 ✓ — consistent.

θ = 22.6° (1 d.p.)

Angles of Elevation & Depression

Real-world trig problems often ask about the angle between a sight line and the horizontal.

Definitions

Angle of elevation — the angle above the horizontal when looking up at something (a building, a kite, a plane).

Angle of depression — the angle below the horizontal when looking down at something (a boat from a cliff, a ground point from a window).

Both Are Measured from the Horizontal

Whether elevation or depression, the angle is measured from the horizontal sight line at the observer's eye, not from the vertical or from the line to the target. The dashed line in the diagrams shows the horizontal reference.

05Worked Example — Angle of Elevation

A flagpole is 15 m tall. An observer stands 20 m horizontally from the base of the flagpole. Find the angle of elevation from the observer to the top of the flagpole. Give your answer to 1 d.p. (3 marks)

MMethod. The observer, the flagpole base, and the flagpole top form a right-angled triangle (right angle at the flagpole base). Relative to the angle of elevation θ at the observer: the 15 m height is opposite, the 20 m horizontal is adjacent. Two legs known → use tan (TOA).

tan θ = opp / adj = 15 / 20

CCalculate.

tan θ = 15/20 = 0.75

θ = tan⁻¹(0.75) (DEG mode)

= 36.869…°

= 36.9° (1 d.p.)

EEvaluate. Sense-check: 15/20 simplifies to 3/4 — the triangle has a 3-4-5 ratio (legs 15 and 20, hypotenuse 25 m). The angle is between 0° and 45° because opposite < adjacent — consistent with 36.9°.

θ = 36.9° (1 d.p.)

Sketching the Right Triangle

Real-world questions don't always come with a diagram — and even when they do, a clean sketch with all sides labelled saves marks.

Routine for word problems

Read the question and identify the right triangle hidden in the scenario

Sketch a clean labelled diagram — mark the right angle, label all known sides and angles, mark the unknown

Choose your reference angle θ; label the three sides as opposite / adjacent / hypotenuse relative to θ

Pick the correct ratio (SOH/CAH/TOA) based on which sides are involved

Substitute, rearrange, and solve — check calculator is in DEG mode

Round to the precision the question asks for; quote with appropriate units

Common Real-World Triangles

Ladder against a wall — angle is between ladder and ground; height is opposite, base is adjacent, ladder is hypotenuse

Ramp / inclined plane — angle of inclination is between ramp and ground; vertical rise is opposite, horizontal run is adjacent

Aeroplane / projectile — flight path is hypotenuse; altitude is opposite the climb angle

Tower viewing — height of tower is opposite the angle of elevation; horizontal distance is adjacent

Cable / rope — typically the hypotenuse, anchored at a height (opposite) and horizontally (adjacent)

06Worked Example — Aeroplane Climb

An aeroplane climbs at a constant angle of 12° above the horizontal. After it has flown 2 km along its flight path, what altitude has it reached? Give your answer to the nearest metre. (4 marks)

MMethod. The flight path is the hypotenuse (2 km = 2000 m). The altitude is the side opposite the 12° angle. Sides involved: opposite and hypotenuse → use sin (SOH).

sin 12° = h / 2000

CCalculate.

Convert flight path to metres: 2 km = 2000 m.

h = 2000 × sin 12°

= 2000 × 0.2079… (DEG mode)

= 415.82… m

= 416 m (nearest metre)

EEvaluate. Sense-check: a small angle (12°) gives a small altitude relative to the flight path — and 416 m is much less than 2000 m ✓. Units: the answer is in metres because we converted km to m before substituting. (Alternative: keep in km — h = 2 × sin 12° = 0.416 km = 416 m, same answer.)

Altitude = 416 m (nearest m)

Calculator Mode & Common Errors

Trig calculations live or die by the calculator setup. Get this right before you substitute any numbers.

Calculator checklist

DEG mode — check the screen displays "DEG" or "D" before any trig button. Radians (RAD or R) gives wildly wrong answers.

Use the inverse keys correctly — sin⁻¹, cos⁻¹, tan⁻¹ via SHIFT + sin/cos/tan when finding an angle

Bracket your fraction — typing "tan⁻¹ 5/12" without brackets may apply tan⁻¹ to 5 only, then divide by 12. Always use tan⁻¹(5/12) with brackets.

Keep full precision — store the calculator value (or use ANS) rather than rounding mid-calculation

Mode-Switch Symptoms

If sin 30° gives you anything other than 0.5, your calculator is not in DEG mode. (In RAD mode, sin 30 — interpreted as 30 radians — gives −0.988.) Always test with sin 30° before starting a question; you should see exactly 0.5.

Sense Checks

For acute angles: sin and cos are between 0 and 1; tan is between 0 and ∞

For sin and cos: as the angle increases from 0° to 90°, sin grows (0→1) and cos shrinks (1→0)

If finding a side, the answer should match the diagram's rough proportions

If finding an angle in a right triangle, the answer must be between 0° and 90° (since the angle is acute)

Examiner Alerts

Trig marks are mostly lost to a small set of recurring errors:

Calculator in Radians Mode

What students get: sin 30° gives −0.988 instead of 0.5. What to write instead: set the calculator to DEG before starting. The screen should show "DEG" or "D".

Wrong Ratio Picked

What students write: "tan θ = opposite / hypotenuse" (mixing up ratios). What to write instead: SOH CAH TOA — tan is opposite over adjacent, not hypotenuse. Identify which two sides are involved before picking the ratio.

Confusing sin with sin⁻¹ (and vice versa)

What students write: Apply sin⁻¹ when looking for a side, or sin when looking for an angle. What to write instead: use sin / cos / tan when finding a SIDE (the angle is known); use sin⁻¹ / cos⁻¹ / tan⁻¹ when finding an ANGLE (two sides are known).

Adjacent Treated as Hypotenuse

What students write: labelling a leg as "hypotenuse" because they mis-orient the triangle. What to write instead: the hypotenuse is always opposite the right angle and is always the longest side. The two sides forming the right angle are the legs (one will be opposite, one adjacent, depending on which non-right angle you choose).

Rounding Mid-Working

What students write: sin 35° = 0.57, then 10 × 0.57 = 5.7 cm. What to write instead: keep the full calculator value (0.5736…), multiply, then round at the end: 10 × 0.5736… = 5.736… → 5.74 cm. Early rounding loses 1-2 marks of precision.

Lesson Checkpoint

You should now be able to:

The Three Trig Ratios (memorise — not on the formula sheet)

sin θ = opposite / hypotenuse (SOH)

cos θ = adjacent / hypotenuse (CAH)

tan θ = opposite / adjacent (TOA)

Decision routine

Identify the right angle and chosen angle θ

Label sides relative to θ: hypotenuse (opposite right angle), opposite (across from θ), adjacent (next to θ, not hyp)

Pick the ratio that uses your two involved sides (one given, one unknown)

Substitute, rearrange, solve — keep full precision until the final round

To find a SIDE: use sin / cos / tan with a known angle

To find an ANGLE: use sin⁻¹ / cos⁻¹ / tan⁻¹ with two known sides

Discipline

Calculator in DEG mode before any trig calculation

Round only at the end, to the precision the question asks for

Check angle of elevation/depression is measured from the horizontal

Sketch the right triangle and label all three sides before substituting

Up Next

Lesson 4.17 covers 3D Pythagoras and Trigonometry (Higher tier only) — extending these techniques to space diagonals, angles between lines and planes, and angles between two planes.