Efficiently Solve the Triangle Using the Law of Sines

Substitute values

3/sin 55^(∘)=c/sin 60^(∘)

▼

Solve for c

LHS * sin 60^(∘)=RHS* sin 60^(∘)

3/sin 55^(∘) (sin 60^(∘))=c

RearrangeEqn

Rearrange equation

c=3/sin 55^(∘) (sin 60^(∘))

Use a calculator

c=3.171665...

RoundSigDig

Round to 3 significant digit(s)

c≈ 3.17

It was found that the distance between the golf course and Magdalena's house is 3.17 kilometers, rounded to three significant figures.

Pop Quiz

Practice Finding Sides Using the Law of Sines

In the following applet, x represents the side length of a triangle. By using the Law of Sines and, if needed, the Triangle Angle Sum Theorem, find the value of x. Write the answer rounded to two decimal places.

Example

Using the Law of Sines To Find Angles of a Triangle

The Law of Sines can also be used to find angle measures of a triangle.

Emily will go backpacking across South America! She will visit Buenos Aires, Santiago, and Asunción, among other cities. Emily knows that the distances from Buenos Aires to Santiago and Asunción are 1140 and 1040 kilometers, respectively. She also knows that the angle whose vertex is located at Santiago and whose sides pass through Buenos Aires and Asunción measures 44^(∘).

To optimize her journey, she wants to find the measure of the remaining two angles of the triangle formed by these three cities, as well as the distance between Santiago and Asunción . Help Emily find them! Write the answers to the nearest integer.

Hint

Start by labeling the sides and the angles of the triangle.

Solution

The angles of the triangle will be labeled A, B, and S. Similarly, the sides will be labeled a, b, and s.

By the Law of Sines, the ratio of the sine of an angle to the length of its opposite side is constant. With this information, a proportion that relates A, a, S, and s can be written. sin S/s=sin A/a In the above formula, a=1140, s=1040, and S=44^(∘) can be substituted. Then the resulting equation can be solved for A, the measure of the angle formed at Asunción.

sin S/s=sin A/a

Substitute values

sin 44^(∘)/1040=sin A/1140

▼

Solve for A

LHS * 1140=RHS* 1140

sin 44^(∘)/1040(1140)=sin A

RearrangeEqn

Rearrange equation

sin A = sin 44^(∘)/1040(1140)

sin^(-1)(LHS) = sin^(-1)(RHS)

A=sin ^(- 1)(sin 44^(∘)/1040(1140))

Use a calculator

A=49.592410... ^(∘)

Round to nearest integer

A≈ 50^(∘)

Rounded to the nearest degree, it was found that the angle formed at Asunción measures 50^(∘).

Finally, to find the measure of the angle at Buenos Aires, the Triangle Angle Sum Theorem can be used. 44^(∘)+ 50^(∘)+B=180^(∘) ⇕ B= 86^(∘) This information can be added to the diagram.

Finally, the Law of Sines can be used again to find the distance between Santiago and Asunción. To do so, recall that the ratio of the length of a side to the sine of its opposite angle is constant. b/sin B=s/sin S Next, s=1040, S=44^(∘), and B=86^(∘) will be substituted into the above equation.

b/sin B=s/sin S

Substitute values

b/sin 86^(∘)=1040/sin 44^(∘)

▼

Solve for b

LHS * sin 86^(∘)=RHS* sin 86^(∘)

b=1040/sin 44^(∘) (sin 86^(∘))

Use a calculator

b=1493.491846...

Round to nearest integer

b≈ 1493

Emily has all she needs to have a great journey!

Pop Quiz

Practice Finding Angles Using the Law of Sines

In the following applet, x represents the measure of an angle of a triangle. By using the Law of Sines and, if needed, the Triangle Angle Sum Theorem, find the value of x. Write the answer as a single number rounded to the nearest degree, without the degree symbol.

Example

Using the Law of Sines To Find Measurements of a Triangle

The Law of Sines can be used to find missing side lengths and angle measures of a triangle. When those side lengths and angle measures are known, the area and perimeter of a triangle can also be found.

Emily is planning to continue her travels. This time, she wants to visit some cities in the UK, as well as Ireland. She would really like to visit London, Edinburgh, and Dublin. It becomes clear to her that these three cities form a triangle. She is able to figure out that the angles at London and Edinburgh measure 41^(∘) and 59^(∘), respectively. She also knows that the distance between these two cities is 530 kilometers.

To optimize her journey, Emily wants to find the perimeter and area of the triangle. Help her with her plan by calculating the perimeter and area! Write the perimeter rounded to the nearest integer and the area rounded to the nearest thousand.

Hint

The perimeter of a triangle is the sum of its three side lengths. The area of the triangle can be found by calculating half the product of two side lengths and the sine of their included angle.

Solution

The perimeter and the area of the triangle will be calculated one at a time.

Perimeter

The perimeter of a triangle is the sum of its three side lengths. Therefore, the missing side lengths need to be found. To make things clearer, the angles and sides will first be labeled.

Note that the missing angle is opposite to the side of the triangle whose length is known. Therefore, in order to use the Law of Sines to find the missing side lengths, the angle at Dublin must be calculated. To do this, the Triangle Angle Sum Theorem can be used. D+ 59^(∘)+ 41^(∘)=180^(∘) ⇕ D= 80^(∘) The measure of the angle at Dublin is 80^(∘).

The Law of Sines can now be used to find the missing side lengths. The law states that, for every triangle, the ratio of the length of a side to the sine of its opposite angle is constant. With this rule in mind, a proportion involving D, d, L and l can be written. l/sin L=d/sin D Next, L= 41^(∘), D= 80^(∘), and d= 530 can be substituted in the above equation, and l can be isolated.

l/sin L=d/sin D

Substitute values

l/sin 41^(∘)=530/sin 80^(∘)

▼

Solve for l

LHS * sin 41^(∘)=RHS* sin 41^(∘)

l = 530/sin 80^(∘)(sin 41^(∘))

Use a calculator

l=353.075292...

Round to nearest integer

l≈ 353

The distance between Dublin and Edinburgh is about 353 kilometers. By following the same procedure, the distance between London and Dublin can be found.

Law of Sines	Substitute	Simplify
l/sin L=d/sin D	l/sin 41^(∘)=530/sin 80^(∘)	l≈ 353
e/sin E=d/sin D	e/sin 59^(∘)=530/sin 80^(∘)	e≈ 461

The distance between London and Dublin is about 461 kilometers.

The perimeter of the triangle formed by the three cities can now be calculated. Perimeter P= 530+ 353+ 461 ⇕ P=1344km

Area

The area of a triangle can be found by calculating half the product of two side lengths and the sine of their included angle. Since all angle measures and sides lengths are known, any of them can be used. In this case, E, d, and l will be arbitrarily used. Area: 1/2 d l sin E The respective values can be substituted to find the area of the triangle.

Area= 1/2 d l sin E

Substitute values

Area= 1/2( 530)( 353) sin 59^(∘)

▼

Evaluate right-hand side

Multiply

Area= 1/2(187 090) sin 59^(∘)

MoveRightFacToNumOne

1/b* a = a/b

Area= 187 090/2 sin 59^(∘)

CalcQuot

Calculate quotient

Area= 93 545 sin 59^(∘)

Use a calculator

Area= 80 183.715144...

Approximate to the nearest thousand

Area≈ 80 000

The area of the triangle formed by London, Dublin, and Edinburgh is about 80 000 square kilometers. Thanks for helping Emily plan her travels!

Discussion

The Ambiguous Case of the Law of Sines

The Law of Sines is useful to find missing angle measures and side lengths for any type of triangle.

Consider a triangle with vertices A, B, and C, where the measure of the angle at C is 34^(∘), and the lengths of AB and BC are 6 and 10 centimeters, respectively. The following diagram represents the triangle that was just described.

The measure of the angle at A can be found by using the Law of Sines.

sin A/10=sin 34^(∘)/6

▼

Solve for A

LHS * 10=RHS* 10

sin A=sin 34^(∘)/6 (10)

CommutativePropMult

Commutative Property of Multiplication

sin A=10 (sin 34^(∘)/6)

MoveLeftFacToNum

a*b/c= a* b/c

sin A=10sin 34^(∘)/6

ReduceFrac

a/b=.a /2./.b /2.

sin A=5sin 34^(∘)/3

sin^(-1)(LHS) = sin^(-1)(RHS)

A=sin ^(- 1)(5sin 34^(∘)/3)

Use a calculator

A=68.746886... ^(∘)

Round to nearest integer

A≈ 69^(∘)

Using the Law of Sines, it was found that the measure of the angle at A is about 69^(∘).

Is this the only possible measure for the angle at A? To find the measure of this angle, the Law of Sines was applied. That is, for any triangle, the ratio of the sine of an angle to the length of its opposite side is constant. sin 69^(∘)/10=sin 34^(∘)/6 Recall now that the sine of an angle is equal to the sine of the supplement of the angle. sin θ = sin (180^(∘)-θ) With this information, it can be stated that the sine of an angle whose measure is 69^(∘) is equal to the sine of an angle whose measure is 180^(∘)- 69^(∘)= 111^(∘). Therefore, by the Substitution Property of Equality, a new proportion can be derived. sin 69^(∘)/10=sin 34^(∘)/6 sin 69^(∘)= sin 111^(∘) ⇓ sin 111^(∘)/10=sin 34^(∘)/6 The obtained proportion implies a new triangle that satisfies the initial conditions.

Therefore, in the given context, there are two valid measures for the angle at A, which are 69^(∘) and 111^(∘). Consequently, the measure of a missing angle is ambiguous. This shows a limitation of the Law of Sines. To avoid this ambiguity, it must be said whether the missing angle is acute or obtuse, or a diagram of the triangle should be provided.

Closure

Calculating an Angle of a Triangle Using the Law of Sines

The challenge presented at the beginning of this lesson can be solved by using the Law of Sines.

Find the measure of the angle at B. Write the answer rounded to the nearest degree.

Hint

In a triangle, the ratio of the sine of an angle to the length of its opposite side is constant.

Solution

Since the ratio of the sine of an angle to the length of its opposite side is constant, a proportion can be written. sin B/AC=sin A/BC ⇒ sin B/15=sin 70^(∘)/20 The above equation can be solved for B.

sin B/15=sin 70^(∘)/20

▼

Solve for B

LHS * 15=RHS* 15

sin B=sin 70^(∘)/20(15)

sin^(-1)(LHS) = sin^(-1)(RHS)

B=sin ^(- 1)(sin 70^(∘)/20(15))

Use a calculator

B=44.810922... ^(∘)