Law of Sines

Cards · 52

Standard formula for the Law of Sines

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

Reciprocal form of the Law of Sines

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

What triangle types does the Law of Sines apply to?

It applies to any planar triangle, both right-angled and oblique (non-right).

What is an oblique triangle?

A triangle that does not contain a right (90-degree) angle.

Primary application: AAS case

Used when two angles and a non-included side are known to solve the remaining parts.

Primary application: ASA case

Used when two angles and the included side are known; find the third angle first, then use the law.

Primary application: SSA case

Used when two sides and a non-included angle are known; leads to the ambiguous case.

Why can the Law of Sines NOT directly solve SAS?

Because no opposite angle-side pair is initially known; use the Law of Cosines first.

Why can the Law of Sines NOT directly solve SSS?

No angles are known initially, so no ratio can be calculated; use the Law of Cosines first.

What is the 'ambiguous case' of the Law of Sines?

The SSA scenario where given values may yield zero, one, or two possible triangles.

Relation of Law of Sines to circumscribed circle diameter

a / sin(A) = b / sin(B) = c / sin(C) = 2R, where R is the circumradius.

Definition of circumradius R

The radius of the circle that passes through all three vertices of the triangle.

Condition for no triangle in SSA when angle A is acute

No triangle exists if side a is strictly less than the altitude h = b * sin(A).

Condition for one right triangle in SSA when angle A is acute

Exactly one right triangle exists if side a equals the altitude h = b * sin(A).

Condition for two triangles in SSA when angle A is acute

Two triangles exist if b * sin(A) < a < b.

Condition for one triangle in SSA when angle A is acute and a >= b

Exactly one triangle exists because side a is greater than or equal to adjacent side b.

Condition for SSA when angle A is obtuse and a <= b

No triangle exists because the side opposite the obtuse angle must be the longest side.

Condition for SSA when angle A is obtuse and a > b

Exactly one triangle exists.

How to compute the second possible angle B2 in SSA

B2 = 180 degrees - B1, provided that A + B2 < 180 degrees.

Why can sin(B) have two values in the interval (0, 180 degrees)?

Because the sine function is positive in both the first and second quadrants: sin(B) = sin(180 - B).

First step to solve an ASA triangle using Law of Sines

Calculate the third angle using the angle sum property: C = 180 - (A + B).

Triangle area formula related to Law of Sines

Area = (1/2) * a * b * sin(C) = (1/2) * b * c * sin(A) = (1/2) * a * c * sin(B).

Area of a triangle in terms of circumradius R and Law of Sines

Area = (a * b * c) / (4R).

Finding side 'a' using Law of Sines given side 'b' and angles A, B

a = (b * sin(A)) / sin(B)

Finding angle 'A' using Law of Sines given sides a, b and angle B

sin(A) = (a * sin(B)) / b, then A = arcsin((a * sin(B)) / b).

What happens if sin(B) > 1 when calculating an angle?

No valid angle exists, meaning no triangle can be formed with the given measurements.

Small angle approximation of Law of Sines

For very small angles measured in radians, a / A ≈ b / B ≈ c / C.

What fundamental geometric theorem ensures angle sum is 180 degrees?

The Triangle Angle Sum Theorem, essential for finding third angles in Law of Sines problems.

Can the Law of Sines be applied in spherical geometry?

Yes, as the Spherical Law of Sines: sin(a)/sin(A) = sin(b)/sin(B) = sin(c)/sin(C).

Is Law of Sines sufficient to solve a triangle given AAA?

No, AAA determines triangle shape and similarity, but cannot determine absolute side lengths.

Role of altitude in deriving the Law of Sines

Dropping an altitude h yields h = b*sin(A) and h = a*sin(B), proving b*sin(A) = a*sin(B).

Solving for c given a, A, and C

c = a * sin(C) / sin(A)

Why prefer reciprocal form sin(A)/a when solving for an angle?

It places the unknown angle in the numerator, simplifying algebraic manipulation.

Why prefer standard form a/sin(A) when solving for a side?

It places the unknown side length in the numerator, simplifying algebraic isolation.

Value of a/sin(90 degrees) in a right triangle with hypotenuse c

Since sin(90) = 1, c / sin(90) = c = 2R, matching basic right-triangle trigonometry.

Effect of rounding intermediate angles during Law of Sines calculation

Premature rounding causes propagation errors; always carry full precision until the final answer.

Law of Sines relationship to the Extended Law of Sines

Extended Law includes the equality to 2R, connecting triangle sides to its circumcircle.

Definition of oblique triangle cases solvable by Law of Sines alone

AAS and ASA always yield a single unique solution; SSA may yield zero, one, or two.

If angle A = 30 deg, side b = 10, what is the altitude to side c?

Altitude h = b * sin(A) = 10 * sin(30 deg) = 5.

In SSA, if a < b * sin(A), how many solutions?

Zero solutions; the side opposite angle A is too short to close the triangle.

In SSA, if a = b * sin(A), how many solutions?

Exactly one solution, forming a right-angled triangle.

In SSA with acute angle A, if a > b, how many solutions?

Exactly one unique solution.

In SSA with acute angle A, if b * sin(A) < a < b, how many solutions?

Two distinct valid solutions (the ambiguous case).

If sin(A) = 0.5, what are the possible values for A in [0, 180]?

A can be 30 degrees or 150 degrees.

How to verify whether the obtuse second angle B2 is valid

Check if the sum of given angle A and B2 is strictly less than 180 degrees.

What is the circumcircle of a triangle?

The unique circle that passes through all three vertices of the triangle.

Ratio comparison: longest side and largest angle

In any triangle, the longest side is always opposite the angle with the largest sine value.

Vector form of triangle area that yields Law of Sines

Magnitude of cross product: |u x v| = |u||v|sin(theta), equal to twice the triangle area.

Navigation use of Law of Sines

Used in triangulation to compute distances to a remote landmark from two known vantage points.

Surveying use of Law of Sines

Determines inaccessible heights and distances by measuring base baselines and elevation angles.

Checklist: Can Law of Sines solve with 3 known sides?

No; must use the Law of Cosines to find at least one angle first.

Checklist: Can Law of Sines solve with 2 sides and included angle?

No; must use Law of Cosines to find the opposite side first.