SOH CAH TOA, special triangles, inverse trig, Law of Sines, and Law of Cosines — with worked examples for every formula.
SOH
Sine = Opposite / Hypotenuse
sin θ = opp/hyp
sin 30° = 1/2
CAH
Cosine = Adjacent / Hypotenuse
cos θ = adj/hyp
cos 60° = 1/2
TOA
Tangent = Opposite / Adjacent
tan θ = opp/adj
tan 45° = 1
The Six Trig Functions
sin θ = opp/hyp
cos θ = adj/hyp
tan θ = opp/adj
csc θ = hyp/opp
sec θ = hyp/adj
cot θ = adj/opp
Sides: 1 : √3 : 2
(short : long : hypotenuse)
sin 30° = 1/2, cos 30° = √3/2
sin 60° = √3/2, cos 60° = 1/2
tan 30° = 1/√3 = √3/3
tan 60° = √3
Memory: the shortest leg (1) is opposite the smallest angle (30°). Hypotenuse is always double the shortest leg.
Sides: 1 : 1 : √2
(leg : leg : hypotenuse)
sin 45° = cos 45° = √2/2 = 1/√2
tan 45° = 1
hyp = leg × √2
leg = hyp / √2 = hyp × √2/2
Memory: a square cut diagonally makes two 45-45-90 triangles. If the side is 1, the diagonal is √2.
Example 1: Given an angle and a side
Right triangle, angle A = 35°, hypotenuse = 12. Find opposite side.
sin 35° = opposite / 12
opposite = 12 · sin 35° = 12 × 0.5736 ≈ 6.88
Example 2: Given two sides, find the angle
Right triangle, opposite = 5, adjacent = 8. Find angle θ.
tan θ = 5/8 = 0.625
θ = arctan(0.625) ≈ 32°
Use inverse trig (tan⁻¹ on your calculator)
Example 3: Angle of elevation
A ladder 10 ft long leans against a wall at 65° from the ground. How high does it reach?
sin 65° = height / 10
height = 10 · sin 65° ≈ 9.06 ft
a / sin A = b / sin B = c / sin C
Side / sin(opposite angle) = constant for all three pairs
Use Law of Sines when you have:
AAS
Two angles + non-included side
ASA
Two angles + included side
SSA
Two sides + non-included angle (ambiguous — check for 0, 1, or 2 solutions)
The Ambiguous Case (SSA)
Given sides a, b and angle A (opposite a): if a < b·sin A → no solution; if a = b·sin A → one right triangle; if b·sin A < a < b → two possible triangles; if a ≥ b → one solution. Always check both possibilities when SSA gives two solutions.
Example (AAS)
A = 30°, B = 70°, a = 8. Find b.
C = 180° − 30° − 70° = 80°
a/sin A = b/sin B → 8/sin 30° = b/sin 70°
b = 8 · sin 70° / sin 30° = 8 × 0.9397 / 0.5 ≈ 15.04
c² = a² + b² − 2ab·cos C
Also: a² = b² + c² − 2bc·cos A
Also: b² = a² + c² − 2ac·cos B
Use Law of Cosines when you have:
SSS
All three sides known, find any angle
SAS
Two sides + included angle, find the third side
Example (SAS)
a = 7, b = 10, C = 50°. Find c.
c² = 7² + 10² − 2(7)(10)·cos 50°
c² = 49 + 100 − 140 × 0.6428 = 149 − 89.99 = 59.01
c = √59.01 ≈ 7.68
Area = ½ bh
Base × Height (right triangles and when height is known)
Area = ½ ab·sin C
Two sides and included angle (SAS case)
Heron's Formula: s = (a+b+c)/2 Area = √[s(s−a)(s−b)(s−c)]
All three sides known (SSS case)
SOH CAH TOA is a memory device for right triangle trig: SOH = Sine = Opposite/Hypotenuse; CAH = Cosine = Adjacent/Hypotenuse; TOA = Tangent = Opposite/Adjacent. The hypotenuse is always the side opposite the right angle (the longest side). Opposite and adjacent are relative to the angle you're working with.
Use Law of Sines (a/sinA = b/sinB = c/sinC) when you have AAS, ASA, or SSA (ambiguous case). Use Law of Cosines (c² = a² + b² − 2ab·cosC) when you have SSS or SAS. If you have a right triangle, SOH CAH TOA is fastest. Law of Cosines is also used when Law of Sines fails (ambiguous case with no solution).
30-60-90 triangle: sides are in ratio 1 : √3 : 2 (short leg : long leg : hypotenuse). The short leg is opposite 30°, long leg opposite 60°. 45-45-90 triangle: sides are in ratio 1 : 1 : √2. Both legs are equal; hypotenuse = leg × √2. These triangles appear on every trig exam.
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