SOH-CAH-TOA only works in right-angled triangles. For any triangle, three rules cover every situation: the sine rule (when you have an angle paired with its opposite side), the cosine rule (when you have two sides + the angle between them, OR all three sides), and the area formula (1/2)ab sin C. A short flowchart tells you which to pick.
Angles in a triangle sum to 180°. If two angles are known, the third comes free — sometimes that’s all you need to unlock the sine rule.
Worked examples
WE 1Sine rule — find a missing side
A triangular sail has angles A = 40° and B = 75°. The side opposite angle A measures 12 m. Find the side opposite angle B.
Identify: opposite pair given (A and a = 12) → sine rule
a/sin A = b/sin B
Substitute
12/sin(40°) = b/sin(75°)
Rearrange for b
b = 12 sin(75°) / sin(40°)
≈ 12 × 0.9659 / 0.6428
≈ 18.0 m (3 sf)
b ≈ 18.0 m
put the UNKNOWN on top: b/sin B = … . That way one rearrangement step gives the answer. GDC in DEG mode.
WE 2Sine rule — find a missing angle
A triangular garden bed PQR has QR = 10 m, PR = 14 m, and angle Q = 85°. Find angle P.
Label: side opposite Q is q = PR = 14 ✓ pair given
side opposite P is p = QR = 10
Use sine rule with sin on top (finding angle)
sin P / p = sin Q / q
sin P / 10 = sin(85°) / 14
Rearrange and apply arcsin
sin P = 10 sin(85°) / 14 ≈ 0.7116
P = sin⁻¹(0.7116) ≈ 45.4° (3 sf)
P ≈ 45.4°
remember to identify the OPPOSITE side for each angle by label. PR is opposite Q (not P).
WE 3Cosine rule — find a missing side (SAS)
Two trails leave a campsite at an angle of 60° to each other. One trail is 5 km long, the other is 8 km long. Find the straight-line distance between the two endpoints of the trails.
Identify: SAS — two sides and the included angle → cosine rule
a = 5, b = 8, C = 60°, find c
Apply cosine rule
c² = a² + b² − 2ab cos C
c² = 5² + 8² − 2(5)(8) cos(60°)
= 25 + 64 − 80 × 0.5
= 89 − 40 = 49
c = √49 = 7 km
distance = 7 km
cos(60°) = 0.5 exactly — a “special angle”. Combined with sides 5 and 8 gives the clean answer 7. (5, 8, 7) is a 60°-triangle pattern.
WE 4Cosine rule — find the largest angle (SSS)
A triangular plot has sides of length 5 m, 8 m and 11 m. Find the largest interior angle.
The largest angle is opposite the longest side
a = 5, b = 8, c = 11 → find C (opposite 11)
Use rearranged cosine rule
cos C = (a² + b² − c²) / (2ab)
= (25 + 64 − 121) / (2 × 5 × 8)
= −32 / 80 = −0.4
Apply arccos
C = cos⁻¹(−0.4) ≈ 113.6° (3 sf)
largest angle ≈ 113.6°
a NEGATIVE value of cos C means the angle is OBTUSE (between 90° and 180°). Always plausible when the side opposite is “too long” for a sharper triangle.
WE 5Area formula — basic SAS
A triangular flag has two sides of length 8 cm and 10 cm meeting at an angle of 30°. Find the area of the flag.
Identify: two sides + included angle → area formula
a = 8, b = 10, C = 30°
Apply A = ½ ab sin C
A = ½ × 8 × 10 × sin(30°)
= ½ × 80 × 0.5
= 20 cm²
area = 20 cm²
sin(30°) = 0.5 exactly — another “special angle” giving a whole-number area. If the angle isn’t between the two given sides, switch to sine or cosine rule first.
WE 6Reverse area formula — find the angle
In triangle ABC, BC = 9 cm, CA = 7 cm and the area of the triangle is 22 cm². Find the angle C.
C is the angle between BC and CA → use area formula
A = ½ × BC × CA × sin C
Substitute knowns
22 = ½ × 9 × 7 × sin C
22 = 31.5 × sin C
Rearrange and apply arcsin
sin C = 22 / 31.5 ≈ 0.6984
C = sin⁻¹(0.6984) ≈ 44.3° (3 sf)
C ≈ 44.3°
the area formula works both ways: find area given an angle, OR find an angle given area + two sides. Same equation, just rearranged.
💡 Top tips
- Always sketch and label first: capitals for angles, lowercase for opposite sides. Half the work is identifying which letter is which.
- Walk the flowchart: right-angled? opposite pair? area? Cosine fills the gap. Don’t try to memorise — let the logic choose.
- For sine rule: orient the formula so the unknown is on top. Side on top → use a/sinA form. Angle on top → flip to sinA/a.
- Cosine rule for angles: a negative cos value = obtuse angle. Don’t panic when the value is below zero.
- Special angles: sin 30° = cos 60° = 0.5; sin 60° = cos 30° = √3/2; sin 45° = cos 45° = √2/2. These often appear in clean-answer problems.
⚠ Common mistakes
- Using SOH-CAH-TOA in a non-right triangle: only works when one angle is 90°. Otherwise pick from sine / cosine / area.
- Wrong pairing in sine rule: a must be opposite A. Sketch with arrows from each angle to its opposite side — eliminates the pairing trap.
- Using the area formula with an angle NOT between the two sides: C in (1/2)ab sinC is the INCLUDED angle. Otherwise find the right pair first.
- GDC in radian mode: AI SL works in degrees. Confirm DEG before any sin / cos / tan calculation.
- Forgetting to square-root after cosine rule: c² = 49 means c = 7, not 49.
Up next: Angles of Elevation & Depression. Apply right-angled trig to real situations: looking up at the top of a tower (elevation), looking down at a boat from a cliff (depression). The tangent ratio does most of the work, and clear diagrams are essential.
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