How this instrument works
The Law of Sines states that in any triangle, the ratio of a side's length to the sine of its OPPOSITE angle is the same constant value for all three sides at once: a ⁄ sinA = b ⁄ sinB = c ⁄ sinC. Once one full side-angle pair is known and one more angle is given, that shared constant can be used to solve for the side opposite the second angle directly: b = a·sinB ⁄ sinA.
This relationship holds regardless of the triangle's shape — acute, obtuse, or right — because it follows from a triangle's own area formula computed three different ways (½ab·sinC and its two rotations), which forces all three ratios to agree. It's the natural tool whenever a triangle problem hands over angles more readily than sides, such as in surveying and navigation, where angles are often easier to measure directly than long distances.
In a right triangle, the Law of Sines still applies and simplifies to exactly what ordinary SOH-CAH-TOA trigonometry would already give, since the side opposite the 90° angle is the hypotenuse and sin(90°) = 1 — the general law and the simpler right-triangle-only shortcut agreeing completely in that special case.
- Enter a known side into the Known side a field.
- Enter that side's opposite angle into the Angle A (opposite side a) field.
- Enter the angle opposite the side you want to find into the Angle B (opposite side b) field.
- Read Side b: the sheet applies a·sinB ⁄ sinA directly.
Worked example — side 10 opposite 30°, second angle 60°
A triangle has a known side a = 10 opposite a 30° angle, and a second angle B = 60°. The side opposite B is b = 10·sin(60°) ⁄ sin(30°) ≈ 17.32 — the ratio side-to-opposite-sine stays fixed across the whole triangle, so once it's known from the first pair, applying it to the second angle recovers the matching side.
With a right triangle instead, a = 10 as the hypotenuse opposite a 90° angle, and a second angle B = 45°: b = 10·sin(45°) ⁄ sin(90°) ≈ 7.07, exactly what plain right-triangle trigonometry (opposite = hypotenuse × sine of the angle) would already give directly.
Questions
What is the Law of Sines?
a ⁄ sinA = b ⁄ sinB = c ⁄ sinC — in any triangle, the ratio of a side's length to the sine of its opposite angle is the same constant for all three sides. Knowing one full side-angle pair lets that constant solve for any other side, given its opposite angle.
When should I use the Law of Sines instead of the Law of Cosines?
Use the Law of Sines when you know at least one full side-angle pair plus one more angle or side. Use the Law of Cosines instead when you know all three sides, or two sides and the angle directly between them (SAS), situations the Law of Sines alone can't solve.
Does the Law of Sines work for right triangles?
Yes, and it reduces to exactly what ordinary right-triangle trigonometry already gives — the side opposite the 90° angle is the hypotenuse, and since sin(90°) = 1, the ratio simplifies down to the familiar opposite-over-hypotenuse relationship.
Why does an isosceles triangle give equal sides for equal angles?
If two angles in a triangle are equal, the Law of Sines forces their opposite sides to be equal too, since sin(A) = sin(B) makes the two ratios collapse to the same side length — this is exactly why an isosceles triangle's two equal angles sit opposite its two equal sides.
Where does the Law of Sines get used practically?
Surveying and navigation rely on it heavily, since angles (measured with a theodolite or compass) are often far easier to determine accurately than long distances — knowing two angles and one side lets the Law of Sines fill in the rest of a triangle's measurements.