SME
Grade 10 Math · Problem type
Three sides: find an angle
Use this for “The sides are 5, 7 and 9. Find the largest angle.”
Diagram
Three sides fix the triangle; the law of cosines gives each angle.
Law of sines: a/sin A = b/sin B = c/sin C ≈ 9.05
This is an example. Type your own values to start a new problem.
c
2
=
a
2
+
b
2
−
2
×
a
×
b
×
cos(
C
°)
A
Angle A
calculated · °
B
Angle B
calculated · °
Formulas
A
+
B
+
C
=
180
33.5573 + 50.7035 + 95.7392 = 180
a
2
=
b
2
+
c
2
−
2
×
b
×
c
×
cos(
A
°)
5
2
=
7
2
+
9
2
−
2
×
7
×
9
×
cos(33.5573°)
b
2
=
a
2
+
c
2
−
2
×
a
×
c
×
cos(
B
°)
7
2
=
5
2
+
9
2
−
2
×
5
×
9
×
cos(50.7035°)
c
2
=
a
2
+
b
2
−
2
×
a
×
b
×
cos(
C
°)
9
2
=
5
2
+
7
2
−
2
×
5
×
7
×
cos(95.7392°)
a
sin(
A
°)
=
b
sin(
B
°)
5
sin(33.5573°)
=
7
sin(50.7035°)
b
sin(
B
°)
=
c
sin(
C
°)
7
sin(50.7035°)
=
9
sin(95.7392°)
a
sin(
A
°)
=
c
sin(
C
°)
5
sin(33.5573°)
=
9
sin(95.7392°)
Assumptions
•
The sides a, b and c are across from the angles A, B and C.
•
Law of sines: a ÷ sin A = b ÷ sin B = c ÷ sin C. Law of cosines: c² = a² + b² − 2ab cos C.
•
Three sides: turn the law of cosines round, cos C = (a² + b² − c²) ÷ (2ab).
Step-by-step
WE KNOW
a = 5, b = 7, c = 9
FIND
angle A (A), angle B (B), angle C (C)
Step 1 · Find angle A (A)
Use a² = b² + c² − 2 × b × c × cos(A°)
Law
of
cosines
turned
round:
three
sides
give
the
angle
A
.
A
=
cos⁻¹(
b
2
+
c
2
−
a
2
2 ×
b
×
c
)
A
=
cos⁻¹(
7
2
+
9
2
−
5
2
2 × 7 × 9
)
A
=
33.5573°
Step 2 · Find angle B (B)
Use b² = a² + c² − 2 × a × c × cos(B°)
Law
of
cosines
turned
round:
three
sides
give
the
angle
B
.
B
=
cos⁻¹(
a
2
+
c
2
−
b
2
2 ×
a
×
c
)
B
=
cos⁻¹(
5
2
+
9
2
−
7
2
2 × 5 × 9
)
B
=
50.7035°
Step 3 · Find angle C (C)
Use c² = a² + b² − 2 × a × b × cos(C°)
Law
of
cosines
turned
round:
three
sides
give
the
angle
C
.
C
=
cos⁻¹(
a
2
+
b
2
−
c
2
2 ×
a
×
b
)
C
=
cos⁻¹(
5
2
+
7
2
−
9
2
2 × 5 × 7
)
C
=
95.7392°
Check
Put the numbers back in. Both sides should match.
33.5573 + 50.7035 + 95.7392 = 180 ✓
5
2
=
7
2
+
9
2
−
2
×
7
×
9
×
cos(33.5573°)
✓
7
2
=
5
2
+
9
2
−
2
×
5
×
9
×
cos(50.7035°)
✓
9
2
=
5
2
+
7
2
−
2
×
5
×
7
×
cos(95.7392°)
✓
5
sin(33.5573°)
=
7
sin(50.7035°)
✓
7
sin(50.7035°)
=
9
sin(95.7392°)
✓
5
sin(33.5573°)
=
9
sin(95.7392°)
✓
Topic check
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Related lessons
The law of sines and the law of cosines
Main lesson
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Area from two sides and the angle between them
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