The Law of Sines
The Law of Sines is basically the ratio of the sine of an angle in degrees and the length of
the side opposite that angle equal to the sine of another angle and the length of the side
opposite that angle. This formula is used to solve triangles given at least the measure of
one angle and side.
Law of Sines : sin sin sin
a b c
α= β= γ
When solving a triangle, make sure your calculator is in degree mode.
SSA Variations Given two sides and its consecutive angle, there are some cases to keep
in mind when deciding the solution.
Angle α a
h = bsinα
Number of
triangles
Acute 0 < a < h 0
Acute a = h 1
Acute h < a < b 2
Acute a ≥ b 1
Obtuse 0 < a ≤ b 0
Obtuse a > b 1
Example Given the triangle below, solve for the missing parts.
Examining the triangle we see that α is obtuse and that a > b. Therefore, we are solving
for one triangle. Now we can apply the Law of Sines to solve the triangle.
γ
α β
c
b a
123û
γ β
47 cm
23 cm c
Law of Sines states: sin sin sin
a b c
α= β= γ
sin123 sin sin
47 23 c
°= β= γ can be separated by keeping the ratio we do know and set it
equal to the ratio where part is known.
sin123 sin
47 23
° = β
23 sin123 sin
47
⋅ °= β
23sin123 sin
47
° = β
0.4104 = sin β
β = sin-1(0.4104) = 24û
Now that we know two angles of the triangle, we are able to determine the measure of the
third angle since the sum of the angles of any triangle is 180û.
180û – (24û + 123û) = 180û – 147û = 33û
Solving for the length of side c by applying the Law of Sines.
sin123 sin 33
47 c
°= °
c sin123û = 47sin33û
47sin 33
sin123
c = °
°
= 30.5 = 31 cm
Example Solve the triangle(s) with β = 26û, a = 1.8 ft, and b = 1.0 ft.
First draw and label what you know about the triangle. Then you can solve as we did
above.
Β = 24û
γ = 33û
c = 31 cm
26û
α
γ
1.8 ft
1.0 ft c Notice this time we are given β and
not α. Let β represent the angle α.
Examining the triangle we see that α is acute and that h = 1.8sin26û, which is
approximately 0.8. Therefore, we are solving for two triangles because h < a < b. Now
we can apply the Law of Sines to solve the triangle.
Law of Sines states: sin sin sin
a b c
α= β= γ
sin 26 sin sin
1.0 1.8 c
°= α= γ can be separated by keeping the ratio we do know and set it
equal to the ratio where part is known.
sin 26 sin
1.0 1.8
° = α
1.8 sin 26 sin
1.0
⋅ °= α
1.8sin 26 sin
1.0
° = α
sin α = 0.7891
α = sin-1(0.7891) = 52û or α = 180û – 52û = 128û
Now that we know two angles of the triangle, we are able to determine the measure of the
third angle since the sum of the angles of any triangle is 180û.
180û – (52û + 26û) = or 180û – (128û + 26û) =
180û – 78û = 102û 180û – 154û = 26û
Solving for the length of side c on both possible triangles by applying the Law of Sines.
sin 26 sin102
1.0 c
°= ° or sin 26 sin 26
1.0 c
°= °
c sin26û = 1.0sin102û c sin26û = 1.0sin26û
1.0sin102
sin 26
c = °
°
= 2.2 ft 1.0sin 26
sin 26
c = °
°
= 1.0 ft
γ = 102û γ = 26û
c = 2.2 ft c = 1.0 ft
α = 52û α = 128û
Try the following:
1. Solve the triangle(s) with β = 98û, a = 62 meters, and b = 88 meters.
2. Solve the triangle(s) with a = 8 km, b = 10 km, and α = 35û.
3. Solve the triangle(s) with α = 123.2û, a = 101 yd, and b = 152 yd.
Answers: α = 44û, γ = 38û, and c = 55 m ; β = 134û, γ = 11û, and c = 2.7 km or β = 46û,
γ = 99û, and c = 14 km ; no triangle is possible
The Law of Sines is basically the ratio of the sine of an angle in degrees and the length of
the side opposite that angle equal to the sine of another angle and the length of the side
opposite that angle. This formula is used to solve triangles given at least the measure of
one angle and side.
Law of Sines : sin sin sin
a b c
α= β= γ
When solving a triangle, make sure your calculator is in degree mode.
SSA Variations Given two sides and its consecutive angle, there are some cases to keep
in mind when deciding the solution.
Angle α a
h = bsinα
Number of
triangles
Acute 0 < a < h 0
Acute a = h 1
Acute h < a < b 2
Acute a ≥ b 1
Obtuse 0 < a ≤ b 0
Obtuse a > b 1
Example Given the triangle below, solve for the missing parts.
Examining the triangle we see that α is obtuse and that a > b. Therefore, we are solving
for one triangle. Now we can apply the Law of Sines to solve the triangle.
γ
α β
c
b a
123û
γ β
47 cm
23 cm c
Law of Sines states: sin sin sin
a b c
α= β= γ
sin123 sin sin
47 23 c
°= β= γ can be separated by keeping the ratio we do know and set it
equal to the ratio where part is known.
sin123 sin
47 23
° = β
23 sin123 sin
47
⋅ °= β
23sin123 sin
47
° = β
0.4104 = sin β
β = sin-1(0.4104) = 24û
Now that we know two angles of the triangle, we are able to determine the measure of the
third angle since the sum of the angles of any triangle is 180û.
180û – (24û + 123û) = 180û – 147û = 33û
Solving for the length of side c by applying the Law of Sines.
sin123 sin 33
47 c
°= °
c sin123û = 47sin33û
47sin 33
sin123
c = °
°
= 30.5 = 31 cm
Example Solve the triangle(s) with β = 26û, a = 1.8 ft, and b = 1.0 ft.
First draw and label what you know about the triangle. Then you can solve as we did
above.
Β = 24û
γ = 33û
c = 31 cm
26û
α
γ
1.8 ft
1.0 ft c Notice this time we are given β and
not α. Let β represent the angle α.
Examining the triangle we see that α is acute and that h = 1.8sin26û, which is
approximately 0.8. Therefore, we are solving for two triangles because h < a < b. Now
we can apply the Law of Sines to solve the triangle.
Law of Sines states: sin sin sin
a b c
α= β= γ
sin 26 sin sin
1.0 1.8 c
°= α= γ can be separated by keeping the ratio we do know and set it
equal to the ratio where part is known.
sin 26 sin
1.0 1.8
° = α
1.8 sin 26 sin
1.0
⋅ °= α
1.8sin 26 sin
1.0
° = α
sin α = 0.7891
α = sin-1(0.7891) = 52û or α = 180û – 52û = 128û
Now that we know two angles of the triangle, we are able to determine the measure of the
third angle since the sum of the angles of any triangle is 180û.
180û – (52û + 26û) = or 180û – (128û + 26û) =
180û – 78û = 102û 180û – 154û = 26û
Solving for the length of side c on both possible triangles by applying the Law of Sines.
sin 26 sin102
1.0 c
°= ° or sin 26 sin 26
1.0 c
°= °
c sin26û = 1.0sin102û c sin26û = 1.0sin26û
1.0sin102
sin 26
c = °
°
= 2.2 ft 1.0sin 26
sin 26
c = °
°
= 1.0 ft
γ = 102û γ = 26û
c = 2.2 ft c = 1.0 ft
α = 52û α = 128û
Try the following:
1. Solve the triangle(s) with β = 98û, a = 62 meters, and b = 88 meters.
2. Solve the triangle(s) with a = 8 km, b = 10 km, and α = 35û.
3. Solve the triangle(s) with α = 123.2û, a = 101 yd, and b = 152 yd.
Answers: α = 44û, γ = 38û, and c = 55 m ; β = 134û, γ = 11û, and c = 2.7 km or β = 46û,
γ = 99û, and c = 14 km ; no triangle is possible
مشكوووووووووووووووووووورررررررررررررررررررررره ابلة
مشكورة ابلة
thanx ya miss so much
