Law of Sines & Triangles

Cards · 52

Basic formula of the Law of Sines

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

Alternative reciprocal form of the Law of Sines

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

Law of Sines relation to circumradius R

a / sin(A) = b / sin(B) = c / sin(C) = 2R, where R is the radius of the circumscribed circle.

Triangle angle sum theorem

The sum of interior angles in any Euclidean triangle satisfies A + B + C = 180 degrees.

When can the Law of Sines directly solve a triangle?

When you are given AAS (Angle-Angle-Side), ASA (Angle-Side-Angle), or SSA (Side-Side-Angle).

Why cannot the Law of Sines immediately solve SAS or SSS cases?

There is no known side-and-opposite-angle pair to set up a solvable single-variable ratio.

Solving AAS: Step 1 when given A, B, and a

Compute angle C directly using C = 180 - (A + B).

Solving AAS: Step 2 to find side b when given A, B, and a

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

Solving AAS: Step 3 to find side c when given A, B, and a

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

Solving ASA: Step 1 when given A, c, and B

Determine angle C using C = 180 - (A + B).

Solving ASA: Step 2 to find side a given A, c, and B

Apply side formula a = c * sin(A) / sin(C).

Solving ASA: Step 3 to find side b given A, c, and B

Apply side formula b = c * sin(B) / sin(C).

Definition of the Ambiguous Case (SSA)

Given two sides and a non-included acute/obtuse angle, there may be 0, 1, or 2 valid triangles.

SSA with acute angle A: condition for 0 solutions when a < h

If a < b * sin(A), side a is too short to reach the opposite side, yielding no valid triangle.

SSA with acute angle A: condition for exactly 1 right triangle

If a = b * sin(A), exactly one right-angled triangle exists where angle B = 90 degrees.

SSA with acute angle A: condition for 2 valid triangles

If b * sin(A) < a < b, two valid triangles exist (one acute B, one obtuse B).

SSA with acute angle A: condition for 1 unique triangle when a >= b

If a >= b, only one triangle is possible because the obtuse alternative would not close.

SSA with obtuse angle A: condition for 0 solutions

If angle A >= 90 degrees and a <= b, no triangle can be formed.

SSA with obtuse angle A: condition for 1 solution

If angle A >= 90 degrees and a > b, exactly one obtuse triangle is formed.

Finding secondary angle B2 in ambiguous SSA case

Calculate B2 = 180 - B1, then check if A + B2 < 180 degrees.

Area of a triangle using sine formula

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

How to isolate sin(B) in SSA case given a, b, A

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

Significance of sin(B) > 1 during SSA calculation

No real angle exists for sin(B) > 1, proving no triangle can be constructed.

Given A = 30 deg, B = 45 deg, a = 10. Find angle C.

C = 180 - 30 - 45 = 105 degrees.

Given A = 30 deg, B = 45 deg, a = 10. Find side b.

b = 10 * sin(45) / sin(30) = 10 * (sqrt(2)/2) / (1/2) = 10 * sqrt(2) approx 14.14.

Given A = 60 deg, a = 6, b = 2*sqrt(3). Find sin(B).

sin(B) = (2*sqrt(3) * sin(60)) / 6 = (2*sqrt(3) * sqrt(3)/2) / 6 = 3 / 6 = 0.5.

Given sin(B) = 0.5 with A = 60 deg, a = 6, b = 2*sqrt(3). Find B.

B = 30 degrees (B = 150 is rejected because 60 + 150 = 210 > 180).

Given A = 40 deg, B = 60 deg, c = 5. Step 1 to solve.

Calculate angle C: C = 180 - (40 + 60) = 80 degrees.

Given A = 40 deg, B = 60 deg, C = 80 deg, c = 5. Find side a.

a = 5 * sin(40) / sin(80) approx 5 * 0.6428 / 0.9848 approx 3.26.

Given A = 40 deg, B = 60 deg, C = 80 deg, c = 5. Find side b.

b = 5 * sin(60) / sin(80) approx 5 * 0.8660 / 0.9848 approx 4.40.

Given A = 120 deg, a = 4, b = 5. How many triangles exist?

Zero triangles exist because A is obtuse and side a is smaller than side b.

Given A = 30 deg, a = 4, b = 10. How many triangles exist?

Zero, because height h = 10 * sin(30) = 5, and a = 4 < h.

Given A = 30 deg, a = 5, b = 10. How many triangles exist?

Exactly one right triangle, because h = 10 * sin(30) = 5 = a.

Given A = 30 deg, a = 7, b = 10. How many triangles exist?

Two triangles, because h = 5 < a < 10.

Given A = 30 deg, a = 12, b = 10. How many triangles exist?

Exactly one triangle, because a > b.

Altitude h from vertex C to side c in terms of angle A

h = b * sin(A).

Altitude h from vertex C to side c in terms of angle B

h = a * sin(B).

Derivation basis of the Law of Sines

Equating two expressions for triangle altitude: h = b * sin(A) = a * sin(B) gives a/sin(A) = b/sin(B).

Can the Law of Sines be applied to right triangles?

Yes; when C = 90 deg, c/sin(90) = c, reducing the law to basic SOH definitions.

Step to find missing angle when two sides and one opposite angle are given

Calculate sin(B) = (b * sin(A)) / a, then apply arcsin to find angle B.

Why check angle sum after using arcsin in SSA?

To determine whether the obtuse supplement (180 - B) leaves room for a third positive angle.

Given A = 45 deg, B = 45 deg, c = 8. Find angle C.

C = 180 - (45 + 45) = 90 degrees.

Given A = 45 deg, C = 90 deg, c = 8. Find side a.

a = 8 * sin(45) / sin(90) = 8 * (sqrt(2)/2) / 1 = 4 * sqrt(2) approx 5.66.

Relationship between greatest side and greatest angle

In any triangle, the longest side is always opposite the angle with the greatest measure.

Relationship between smallest side and smallest angle

In any triangle, the shortest side is always opposite the angle with the smallest measure.

Circumradius formula from Law of Sines

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

Given a = 14, A = 30 deg. Find the diameter of the circumscribed circle.

Diameter = 2R = a / sin(A) = 14 / sin(30) = 14 / 0.5 = 28.

Given b = 10, B = 90 deg. Find circumradius R.

2R = 10 / sin(90) = 10, so circumradius R = 5.

Given a = 6, b = 8, C = 30 deg. Find triangle area.

Area = 0.5 * 6 * 8 * sin(30) = 24 * 0.5 = 12 square units.

How rounding intermediate angles can introduce error

Rounding an angle early distorts subsequent sine values; store exact values in memory.

Domain limitation of arcsine when solving for angles

Arcsin returns angles in [-90, 90] degrees; triangle angles must fall strictly in (0, 180) degrees.

Role of the Law of Sines in navigation and triangulation

Determines an unknown distance to an object by measuring two angles from a known baseline.