MathematicsTrigonometryA-Level

AQACCEAEdexcelOCRWJECAPIBSAT

Sine rule

Relate sides and angles in any triangle.

Understand the formulaSee the free derivationOpen the full walkthrough

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

Open Full Walkthrough Try Calculator

This public page keeps the free explanation visible and leaves premium worked solving, advanced walkthroughs, and saved study tools inside the app.

Core idea

Overview

The Sine Rule, also known as the Law of Sines, defines the constant ratio between the side lengths of a triangle and the sine of their opposite angles. This relationship applies to all triangles, including non-right oblique triangles, making it a more versatile tool than basic SOH-CAH-TOA ratios.

When to use: Apply this rule when you are given a side and its opposite angle along with at least one other side or angle. It is the primary method for solving Angle-Angle-Side (AAS) and Angle-Side-Angle (ASA) cases, as well as the ambiguous Side-Side-Angle (SSA) case.

Why it matters: This equation is essential for triangulation in land surveying, coastal navigation, and astronomy to determine distances to remote objects. It also allows engineers to calculate stress vectors in non-perpendicular structural frameworks.

Symbols

Variables

a = Side a, A = Angle A, b = Side b, B = Angle B

a

Side a

V a r iab l e

A

Angle A

d e g

b

Side b

V a r iab l e

B

Angle B

d e g

Walkthrough

Derivation

Derivation of the Sine Rule

The sine rule relates the sides of a triangle to the sines of their opposite angles. It can be derived by dropping a perpendicular and using right-angled trigonometry.

Triangle has sides a, b, c opposite angles A, B, C respectively.
Euclidean geometry applies.

Drop a Perpendicular Height:

Drop a perpendicular from C to side c. In the left right-angled triangle, the height is $b sin (A)$ .

h = b sin (A)

Express the Same Height Another Way:

In the right right-angled triangle, the same height is $a sin (B)$ .

h = a sin (B)

Equate the Two Expressions:

Both expressions equal the same height h, so they must be equal.

b sin (A) = a sin (B)

Rearrange to the Sine Rule Form:

Rearranging gives the ratio form. Repeating from a different vertex shows the equality with $\frac{c}{s i n ( C )}$ too.

\frac{a}{sin ( A )} = \frac{b}{sin ( B )} = \frac{c}{sin ( C )}

Result

\frac{a}{sin ( A )} = \frac{b}{sin ( B )} = \frac{c}{sin ( C )}

Source: Standard curriculum — A-Level Pure Mathematics (Trigonometry)

Visual intuition

Graph

The graph is a straight line passing through the origin because side a is directly proportional to side b by the constant factor of sin A divided by sin B. For a student, this linear relationship means that doubling side b will always result in a doubling of side a. Large values of side b correspond to proportionally large values of side a, while small values of side b result in small values of side a. The most important feature is that the constant slope represents the fixed ratio between the sines of the two angl

Graph type: linear

Why it behaves this way

Intuition

Imagine a triangle where each side's length is directly proportional to the 'openness' (represented by the sine) of its opposite angle, maintaining a constant ratio for all three side-angle pairs.

a, b, c

The length of a side of the triangle.

A longer side is opposite a larger angle, and a shorter side is opposite a smaller angle, maintaining a constant ratio across the triangle.

A, B, C

The measure of an interior angle of the triangle, opposite to sides a, b, c respectively.

A larger angle corresponds to a larger sine value (for angles within a triangle), which means its opposite side must be proportionally longer to maintain the constant ratio defined by the rule.

sin A, sin B, sin C

The sine of an interior angle of the triangle.

The sine function acts as a scaling factor for the opposite side length. It ensures that the ratio of any side length to the sine of its opposite angle is consistent for all three side-angle pairs within the triangle.

Free study cues

Insight

Canonical usage

All side lengths (a, b, c) must be expressed in the same unit of length. Angles (A, B, C) are typically used in degrees or radians, with the sine function yielding a dimensionless output.

Common confusion

A common mistake is using inconsistent units for the side lengths within the same problem (e.g., one side in meters, another in centimeters).

Unit systems

$a$ m · Any consistent unit of length (e.g., cm, km, ft) may be used, but all sides must use the same unit.

$b$ m · Any consistent unit of length (e.g., cm, km, ft) may be used, but all sides must use the same unit.

$c$ m · Any consistent unit of length (e.g., cm, km, ft) may be used, but all sides must use the same unit.

$A$ degrees · Angles are typically given in degrees or radians. The sine function itself yields a dimensionless value.

$B$ degrees · Angles are typically given in degrees or radians. The sine function itself yields a dimensionless value.

$C$ degrees · Angles are typically given in degrees or radians. The sine function itself yields a dimensionless value.

One free problem

Practice Problem

In triangle ABC, find the length of side a if angle A is 40°, side b is 10 units, and angle B is 60°.

Angle A40 deg

Side b10

Angle B60 deg

Solve for: $a$

Hint: Rearrange the formula to solve for side a: a = (b × sin A) / sin B.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Triangulation in surveying.

Study smarter

Tips

When solving for an angle, use the reciprocal form sin(A)/a = sin(B)/b to isolate the unknown more easily.
Check the Side-Side-Angle (SSA) condition for the 'ambiguous case' where two different triangles might satisfy the given data.
Ensure your calculator is set to 'Degrees' rather than 'Radians' unless the problem specifies otherwise.
The longest side of a triangle must always be opposite the largest angle.

Avoid these traps

Common Mistakes

Using radians incorrectly.
Inverting one side of the ratio.

Common questions

Frequently Asked Questions

The sine rule relates the sides of a triangle to the sines of their opposite angles. It can be derived by dropping a perpendicular and using right-angled trigonometry.

Apply this rule when you are given a side and its opposite angle along with at least one other side or angle. It is the primary method for solving Angle-Angle-Side (AAS) and Angle-Side-Angle (ASA) cases, as well as the ambiguous Side-Side-Angle (SSA) case.

This equation is essential for triangulation in land surveying, coastal navigation, and astronomy to determine distances to remote objects. It also allows engineers to calculate stress vectors in non-perpendicular structural frameworks.

Using radians incorrectly. Inverting one side of the ratio.

Triangulation in surveying.

When solving for an angle, use the reciprocal form sin(A)/a = sin(B)/b to isolate the unknown more easily. Check the Side-Side-Angle (SSA) condition for the 'ambiguous case' where two different triangles might satisfy the given data. Ensure your calculator is set to 'Degrees' rather than 'Radians' unless the problem specifies otherwise. The longest side of a triangle must always be opposite the largest angle.

References

Sources

Wikipedia: Law of sines
Britannica: Law of sines
Neill, H. (2016). Pure Mathematics 1 (Cambridge International AS & A Level Mathematics). Cambridge University Press.
IUPAC Gold Book, 'dimensionless quantity'
Law of sines Wikipedia article
Spherical trigonometry Wikipedia article
Euclidean geometry Wikipedia article
Degenerate triangle Wikipedia article

Sine rule

Overview

Variables

Derivation

Drop a Perpendicular Height:

Express the Same Height Another Way:

Equate the Two Expressions:

Rearrange to the Sine Rule Form:

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Cosine rule

Frequently Asked Questions

Sources