UNIT 9 • GEOMETRY & TRIGONOMETRY

Triangle Solver
Sine Rule & Cosine Rule

A triangle solver finds every side and angle from three facts, as long as at least one is a side. The cosine rule $a^2 = b^2 + c^2 - 2bc\cos A$ handles SSS and SAS; the sine rule $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ handles the rest.

THE TWO RULESANY TRIANGLE

\begin{gathered} \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \\ a^2 = b^2 + c^2 - 2bc\cos A \end{gathered}

1. LabelSide

a

is opposite angle

A

b

opposite

B

c

opposite

C

2. Choose the RuleSSS or SAS: cosine rule. A side with its opposite angle known: sine rule.

3. Angles Add to 180°Once two angles are known,

C = 180^\circ - A - B

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Solve the triangle with sides $a = 7$ , $b = 8$ , $c = 9$ , and find its area.

STEP 01 • ANGLE A BY THE COSINE RULE

\cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{64 + 81 - 49}{2 \times 8 \times 9} = \frac{96}{144} = \frac{2}{3} \Rightarrow A \approx 48.19^\circ

STEP 02 • ANGLE B, THEN ANGLE C

\cos B = \frac{a^2 + c^2 - b^2}{2ac} = \frac{49 + 81 - 64}{2 \times 7 \times 9} = \frac{66}{126} = \frac{11}{21} \Rightarrow B \approx 58.41^\circ

C = 180^\circ - 48.19^\circ - 58.41^\circ \approx 73.40^\circ

Check with the cosine rule: $\cos C = \frac{49 + 64 - 81}{2 \times 7 \times 8} = \frac{2}{7}$ , which gives $C \approx 73.40^\circ$ ✓

STEP 03 • AREA

Since $\cos A = \tfrac{2}{3}$ , $\sin A = \sqrt{1 - \tfrac{4}{9}} = \tfrac{\sqrt{5}}{3}$ :

\text{Area} = \tfrac{1}{2}bc\sin A = \tfrac{1}{2} \times 8 \times 9 \times \frac{\sqrt{5}}{3} = 12\sqrt{5} \approx 26.83

FINAL CONCLUSION

A ≈ 48.19°, B ≈ 58.41°, C ≈ 73.40°, Area = 12√5

\text{Area} = 12\sqrt{5} \approx 26.83

All three angles are under 90°, so this is an acute scalene triangle. The calculator below opens on it.

LABORATORY

Interactive Triangle Solver

LABORATORY

Triangle Solver Engine

What do you know?

You know three sides. Side a is opposite angle A, b opposite B, c opposite C.

Side aSide bSide c

Common Questions

When do you use the sine rule and when the cosine rule?

Use the cosine rule when you know three sides (SSS) or two sides and the angle between them (SAS). Use the sine rule when you know a side and the angle opposite it, as in ASA, AAS and SSA.

What is the ambiguous case of the sine rule?

With two sides and an angle that is not between them (SSA), the sine rule gives

\sin B

, and two angles have that sine:

B

and

180^\circ - B

. So there can be two different triangles, one, or none. The calculator shows every one that fits.

How do you find the area of a triangle from two sides and an angle?

\text{Area} = \tfrac{1}{2}ab\sin C

: half the product of two sides times the sine of the angle between them.

Why can't I solve a triangle from three angles (AAA)?

Three angles fix the shape but not the size: every enlargement of the triangle has the same angles. You need at least one side.

PreviousPythagoras Theorem

Unit 9•42 of 47

NextUnit Circle

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 9 • GEOMETRY & TRIGONOMETRY

Triangle Solver
Sine Rule & Cosine Rule

THE TWO RULESANY TRIANGLE

\begin{gathered} \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \\ a^2 = b^2 + c^2 - 2bc\cos A \end{gathered}

1. LabelSide

a

is opposite angle

A

b

opposite

B

c

opposite

C

2. Choose the RuleSSS or SAS: cosine rule. A side with its opposite angle known: sine rule.

3. Angles Add to 180°Once two angles are known,

C = 180^\circ - A - B

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Solve the triangle with sides $a = 7$ , $b = 8$ , $c = 9$ , and find its area.

STEP 01 • ANGLE A BY THE COSINE RULE

\cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{64 + 81 - 49}{2 \times 8 \times 9} = \frac{96}{144} = \frac{2}{3} \Rightarrow A \approx 48.19^\circ

STEP 02 • ANGLE B, THEN ANGLE C

\cos B = \frac{a^2 + c^2 - b^2}{2ac} = \frac{49 + 81 - 64}{2 \times 7 \times 9} = \frac{66}{126} = \frac{11}{21} \Rightarrow B \approx 58.41^\circ

C = 180^\circ - 48.19^\circ - 58.41^\circ \approx 73.40^\circ

Check with the cosine rule: $\cos C = \frac{49 + 64 - 81}{2 \times 7 \times 8} = \frac{2}{7}$ , which gives $C \approx 73.40^\circ$ ✓

STEP 03 • AREA

Since $\cos A = \tfrac{2}{3}$ , $\sin A = \sqrt{1 - \tfrac{4}{9}} = \tfrac{\sqrt{5}}{3}$ :

\text{Area} = \tfrac{1}{2}bc\sin A = \tfrac{1}{2} \times 8 \times 9 \times \frac{\sqrt{5}}{3} = 12\sqrt{5} \approx 26.83

FINAL CONCLUSION

A ≈ 48.19°, B ≈ 58.41°, C ≈ 73.40°, Area = 12√5

\text{Area} = 12\sqrt{5} \approx 26.83

All three angles are under 90°, so this is an acute scalene triangle. The calculator below opens on it.

LABORATORY

Interactive Triangle Solver

LABORATORY

Triangle Solver Engine

What do you know?

You know three sides. Side a is opposite angle A, b opposite B, c opposite C.

Side aSide bSide c

Common Questions

When do you use the sine rule and when the cosine rule?

Use the cosine rule when you know three sides (SSS) or two sides and the angle between them (SAS). Use the sine rule when you know a side and the angle opposite it, as in ASA, AAS and SSA.

What is the ambiguous case of the sine rule?

With two sides and an angle that is not between them (SSA), the sine rule gives

\sin B

, and two angles have that sine:

B

and

180^\circ - B

. So there can be two different triangles, one, or none. The calculator shows every one that fits.

How do you find the area of a triangle from two sides and an angle?

\text{Area} = \tfrac{1}{2}ab\sin C

: half the product of two sides times the sine of the angle between them.

Why can't I solve a triangle from three angles (AAA)?

Three angles fix the shape but not the size: every enlargement of the triangle has the same angles. You need at least one side.

PreviousPythagoras Theorem

Unit 9•42 of 47

NextUnit Circle

Triangle Solver Sine Rule & Cosine Rule

Step-by-Step Worked Example

A ≈ 48.19°, B ≈ 58.41°, C ≈ 73.40°, Area = 12√5

Interactive Triangle Solver

Triangle Solver Engine

Common Questions

NETZ Engine

Triangle Solver Sine Rule & Cosine Rule

Step-by-Step Worked Example

A ≈ 48.19°, B ≈ 58.41°, C ≈ 73.40°, Area = 12√5

Interactive Triangle Solver

Triangle Solver Engine

Common Questions

Triangle Solver
Sine Rule & Cosine Rule

Triangle Solver
Sine Rule & Cosine Rule