SME
Grade 10 Math · Geometry
The law of sines and the law of cosines
Use this for “A = 35°, B = 80° and a = 9. Find the other sides” with the law of sines.
Diagram
Two angles and a side fix the triangle; the law of sines gives the other sides.
Law of sines: a/sin A = b/sin B = c/sin C ≈ 15.69
This is an example. Type your own values to start a new problem.
a
sin(
°)
=
b
sin(
°)
c
Side c
calculated
C
Angle C
calculated · °
Formulas
A
+
B
+
C
=
180
35 + 80 + 65 = 180
a
sin(
A
°)
=
b
sin(
B
°)
9
sin(35°)
=
15.4526
sin(80°)
a
sin(
A
°)
=
c
sin(
C
°)
9
sin(35°)
=
14.2209
sin(65°)
b
sin(
B
°)
=
c
sin(
C
°)
15.4526
sin(80°)
=
14.2209
sin(65°)
Assumptions
•
Each side is across from the angle with the same letter.
•
A side divided by the sine of its angle is the same for all three sides.
•
Two angles and any side (AAS or ASA): find the third angle from 180° first.
Step-by-step
WE KNOW
a = 9, A = 35°, B = 80°
FIND
angle C (C), side b (b), side c (c)
Step 1 · Find angle C (C)
Use A + B + C = 180
The angles of a triangle add to 180°.
C
=
180
−
A
−
B
C
=
180
−
35
−
80
C
=
65°
Step 2 · Find side b (b)
Use a ÷ sin(A°) = b ÷ sin(B°)
Law of sines: multiply both sides by the sine of the angle across from the side.
b
=
a
×
sin(
B
°)
sin(
A
°)
b
=
9
×
sin(80°)
sin(35°)
b
=
9
×
0.9848
0.5736
b
=
8.8632
0.5736
b
=
15.4526
Step 3 · Find side c (c)
Use a ÷ sin(A°) = c ÷ sin(C°)
Law of sines: multiply both sides by the sine of the angle across from the side.
c
=
a
×
sin(
C
°)
sin(
A
°)
c
=
9
×
sin(65°)
sin(35°)
c
=
9
×
0.9063
0.5736
c
=
8.1567
0.5736
c
=
14.2209
Check
Put the numbers back in. Both sides should match.
35 + 80 + 65 = 180 ✓
9
sin(35°)
=
15.4526
sin(80°)
✓
9
sin(35°)
=
14.2209
sin(65°)
✓
15.4526
sin(80°)
=
14.2209
sin(65°)
✓
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Sine and cosine of complementary angles
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Two sides and the angle between them
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