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Law of Sines Calculator

Select parameters and fetch their values along with the chosen units. The tool will quickly apply the law of sines to calculate all sides and angles of the triangle

Calculate:

Find a:

cm ▾

Find b:

cm ▾

Find c:

cm ▾

\( A = \sin^{-1} \left[ \dfrac{a \sin B}{b} \right] \)

Law of Sines

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Law of Sines Calculator

An online law of sines calculator helps you find unknown angles and side lengths of a triangle. It applies the law of sines formula to solve simple or complex trigonometric problems efficiently.

What is the Law of Sines?

The law of sines relates the angles and sides of a triangle by stating that the ratio of a side to the sine of its opposite angle is constant:

Law of Sines Calculator

Sides: a, b, c
Angles: A, B, C
Other characteristics: P = perimeter, s = semi-perimeter, K = area, r = inradius, R = circumradius

Formula:

\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

Example

Compute the length of side b:

Law of Sines Example

Solution:

Using the law of sines:

\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]

\[ \frac{12}{\sin 50^\circ} = \frac{b}{\sin 100^\circ} \]

Cross multiply:

\[ b = \frac{12 \cdot \sin 100^\circ}{\sin 50^\circ} \approx 15.427 \]

Thus, side b ≈ 15.427 mm. An online sine calculator can compute sin(x) in degrees, radians, or π radians.

Equations Derived from Law of Sines for Angles

A = sin⁻¹(a sin B / b)
A = sin⁻¹(a sin C / c)
B = sin⁻¹(b sin A / a)
B = sin⁻¹(b sin C / c)
C = sin⁻¹(c sin A / a)
C = sin⁻¹(c sin B / b)

Equations Derived from Law of Sines for Sides

a = b sin A / sin B
a = c sin A / sin C
b = a sin B / sin A
b = c sin B / sin C
c = a sin C / sin A
c = b sin C / sin B

Ambiguous Case in Law of Sines

An ambiguous case occurs when two different triangles can be constructed from the given data. They are labeled ABC and AB'C'.

Ambiguous Case Law of Sines

Conditions for ambiguity:

Two sides and a non-included angle are given (SSA)
Given angle A < 90°
Side a is shorter than side c but longer than the altitude h from angle B (a > h)

The CSC Calculator can calculate cosecant and inverse sine values for given angles.

How the Law of Sines Calculator Works

Input:

Select the side or angle to calculate; the calculator shows the corresponding formula.
Enter known sides and angles.
Select units (degrees, radians, etc.).
Click Calculate.

Output:

Calculated angles and sides
Other triangle characteristics (area, perimeter, inradius, circumradius)
Triangle diagram based on the given values

FAQs

When to use the Law of Sines?

Use it when you know two angles and a side (AAS or ASA) or two sides and a non-included angle (SSA).

Triangle Side Rule

The sum of any two sides must be greater than the third side (Triangle Inequality Theorem).

What is an Oblique Triangle?

An oblique triangle has no right angle. Standard trigonometric ratios are modified using the laws of sines and cosines.

Triangle Characteristics

Circumradius: R = abc / 4K
Inradius: r = √((s-a)(s-b)(s-c)/s)
Area: K = √(s(s-a)(s-b)(s-c))
Semi-perimeter: s = 0.5(a + b + c)
Perimeter: P = a + b + c

Conclusion

The law of sines calculator quickly finds missing sides and angles using the law of sines formula, providing accurate results instantly. Ideal for students, engineers, and anyone working with triangles and trigonometry.

References

Wikipedia: Law of Sines – Ambiguous case, circumcircle, area relations
Dave's Short Trig Course: Oblique Triangles
Khan Academy: Law of Sines Review

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