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Right-triangle trigonometry

SOH CAH TOAsine cosine tangentlaw of sineslaw of cosinessolving triangles

The three ratios for right triangles and the two laws that extend them to any triangle, plus the two-sides-and-an-angle area formula.

Right-Triangle Sine Ratio (SOH)

sin⁡θ=oh\sin\theta = \frac{o}{h}

Relates an acute angle of a right triangle to its opposite side and the hypotenuse.

Right-Triangle Cosine Ratio (CAH)

cos⁡θ=ah\cos\theta = \frac{a}{h}

Relates an acute angle of a right triangle to its adjacent side and the hypotenuse.

Right-Triangle Tangent Ratio (TOA)

tan⁡θ=oa\tan\theta = \frac{o}{a}

Relates an acute angle of a right triangle to the ratio of its opposite and adjacent legs.

Law of Sines

asin⁡A=bsin⁡B\frac{a}{\sin A} = \frac{b}{\sin B}

In any triangle, each side is proportional to the sine of its opposite angle.

Law of Cosines

c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C

Finds the third side of any triangle from two sides and their included angle, or the angle from all three sides.

Triangle Area (Two Sides and Included Angle)

A=12 absin⁡CA = \tfrac{1}{2}\,ab\sin C

Computes a triangle's area from two sides and the angle between them, with no height needed.

How they fit together

SOH CAH TOA only works if there is a right angle. The three ratios each pair the hypotenuse or the two legs, and which one you use is decided entirely by which two sides the problem mentions relative to your angle. For triangles without a right angle you need the two laws, and the law of cosines is really Pythagoras with a correction term: when the angle is 90°, cos C is zero and it collapses back to c² = a² + b².

The decision rule is short. Right triangle: SOH CAH TOA. No right angle, and you have a matched side-and-opposite-angle pair: law of sines. No right angle, and you have three sides, or two sides with the angle between them: law of cosines. Two sides and the included angle also give the area directly, without hunting for a height. The classic trap is the ambiguous case: given two sides and a non-included angle, the law of sines can have two valid solutions, because sin θ = sin(180° − θ), and a calculator's arcsine only ever reports the acute one. Check whether the obtuse partner also closes the triangle. And confirm the calculator's angle mode before anything else — degrees for surveying and construction, radians for calculus.