SME
Grade 10 Math · Problem type
Two sides and the angle between them
Use this for “b = 7, c = 10 and A = 50°. Find a.”
Diagram
Two sides and the angle between them fix the triangle: the law of cosines gives the third side.
Law of sines: a/sin A = b/sin B = c/sin C ≈ 10.03
This is an example. Type your own values to start a new problem.
a
2
=
b
2
+
c
2
−
2
×
b
×
c
×
cos(
A
°)
B
Angle B
calculated · °
C
Angle C
calculated · °
Formulas
A
+
B
+
C
=
180
50 + 44.2712 + 85.7288 = 180
a
2
=
b
2
+
c
2
−
2
×
b
×
c
×
cos(
A
°)
7.6818
2
=
7
2
+
10
2
−
2
×
7
×
10
×
cos(50°)
b
2
=
a
2
+
c
2
−
2
×
a
×
c
×
cos(
B
°)
7
2
=
7.6818
2
+
10
2
−
2
×
7.6818
×
10
×
cos(44.2712°)
c
2
=
a
2
+
b
2
−
2
×
a
×
b
×
cos(
C
°)
10
2
=
7.6818
2
+
7
2
−
2
×
7.6818
×
7
×
cos(85.7288°)
a
sin(
A
°)
=
b
sin(
B
°)
7.6818
sin(50°)
=
7
sin(44.2712°)
b
sin(
B
°)
=
c
sin(
C
°)
7
sin(44.2712°)
=
10
sin(85.7288°)
a
sin(
A
°)
=
c
sin(
C
°)
7.6818
sin(50°)
=
10
sin(85.7288°)
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.
•
Two sides and the angle between them: the law of cosines gives the third side first.
Step-by-step
WE KNOW
b = 7, c = 10, A = 50°
FIND
side a (a), angle B (B), angle C (C)
Step 1 · Find side a (a)
Use a² = b² + c² − 2 × b × c × cos(A°)
Law
of
cosines:
the
two
sides
beside
A
and
the
angle
between
them
give
the
side
across
from
it.
a
=
√(
b
2
+
c
2
−
2
×
b
×
c
×
cos(
A
°))
a
=
√(
7
2
+
10
2
−
2
×
7
×
10
×
cos(50°))
a
=
√
49 + 100 − 2 × 7 × 10 × 0.6428
a
=
√
49 + 100 − 14 × 10 × 0.6428
a
=
√
49 + 100 − 140 × 0.6428
a
=
√
49 + 100 − 89.992
a
=
√
59.008
a
=
7.6818
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(
7.6818
2
+
10
2
−
7
2
2 × 7.6818 × 10
)
B
=
44.2712°
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(
7.6818
2
+
7
2
−
10
2
2 × 7.6818 × 7
)
C
=
85.7288°
Check
Put the numbers back in. Both sides should match.
50 + 44.2712 + 85.7288 = 180 ✓
7.6818
2
=
7
2
+
10
2
−
2
×
7
×
10
×
cos(50°)
✓
7
2
=
7.6818
2
+
10
2
−
2
×
7.6818
×
10
×
cos(44.2712°)
✓
10
2
=
7.6818
2
+
7
2
−
2
×
7.6818
×
7
×
cos(85.7288°)
✓
7.6818
sin(50°)
=
7
sin(44.2712°)
✓
7
sin(44.2712°)
=
10
sin(85.7288°)
✓
7.6818
sin(50°)
=
10
sin(85.7288°)
✓
Topic check
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The law of sines and the law of cosines
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