a Find the measure of ∠B when it is acute. Use this value to find the measure of ∠B when it is obtuse.
B
b Find the measure of ∠B when it is acute. Use this value to find the measure of ∠B when it is obtuse.
A
a
First Triangle:
m∠B≈ 52.3^(∘), m∠C≈ 87.7^(∘), c≈ 20.2
Second Triangle:
m∠B≈ 127.7^(∘), m∠C≈ 12.3^(∘), c≈ 4.3
B
b
First Triangle:
m∠B≈ 42.4^(∘), m∠C≈ 116.6^(∘), c≈ 42.4
Second Triangle:
m∠B≈ 137.6^(∘), m∠C≈ 21.4^(∘), c≈ 17.3
Practice makes perfect
a We are given the measure of one acute angle of â–³ ABC, the length of one adjacent side, and the length of the side opposite that angle, which is less than the length of the adjacent side.
Acute Angle:& &m∠A= 40^(∘)
Opposite Side:& &a= 13
Adjacent Side:& &b= 16
We will use the Law of Sines to solve â–³ ABC. This is an ambiguous case, so there are two possible solutions for â–³ ABC.
Let's find these solutions one at a time.
Solution I
In the first case we assume that ∠B is an acute angle.
Now we will apply the Law of Sines to the above triangle.
13/sin 40^(∘)=16/sinB=c/sinC
The obtained expression can be split into two equations.
13/sin40^(∘)=16/sinB and 13/sin40^(∘)=c/sinC
We will solve the first equation for the measure of ∠B. First, let's rewrite the equation so that the sine of ∠B is isolated on the left-hand side.
Now we will use the inverse sine ratio.
sinB=16sin40^(∘)/13
⇕
m∠B=sin^(- 1)(16sin40^(∘)/13)
Let's approximate the value of m∠B to the nearest tenth of a degree using a calculator.
m∠B=sin^(- 1)(16sin40^(∘)/13)≈ 52.3^(∘)
The measure of ∠B is about 52.3^(∘). We will find the measure of the third angle ∠C using the Triangle Sum Theorem.
m∠C≈ 180^(∘)- 40^(∘)- 52.3^(∘)=87.7^(∘)
The measure of ∠C is about 87.7^(∘). Finally we will use the second equation to estimate the length c of the remaining side.
13/sin40^(∘)≈c/sin 87.7^(∘) ⇔ c≈13sin87.7^(∘)/sin40^(∘)
Let's use a calculator.
c≈13sin87.7^(∘)/sin40^(∘)≈ 20.2
The length c of the remaining side of the triangle is about 20.2 units.
Solution II
In the second case we assume that ∠B is an obtuse angle.
Remember that the inverse sine function gives only acute angle measures. Therefore, in this case we will find m∠B using supplementary angles. In the first solution we have found the measure of the supplement of the current ∠B, about 52.3^(∘).
Now we can find the measures of ∠B and ∠C.
& m∠B≈ 180^(∘)- 52.3^(∘)=127.7^(∘)
& m∠C≈ 180^(∘)- 40^(∘)-127.7^(∘)=12.3^(∘)
Finally, let's find the value of c using the Law of Sines.
13/sin 40^(∘)≈c/sin12.3^(∘) ⇔ c≈ 4.3
The length c of the remaining side of the triangle is about 4.3 units.
b We are given the measure of one acute angle of â–³ ABC, the length of one adjacent side, and the length of the side opposite that angle, which is less than the length of the adjacent side.
Acute Angle:& &m∠A= 21^(∘)
Opposite Side:& &a= 17
Adjacent Side:& &b= 32
We will use the Law of Sines to solve â–³ ABC. This is an ambiguous case, so there are two possible solutions for â–³ ABC.
Let's find these solutions one at a time.
Solution I
In the first case we assume that ∠B is an acute angle.
Now we will apply the Law of Sines to the above triangle.
17/sin 21^(∘)=32/sinB=c/sinC
The obtained expression can be split into two equations.
17/sin21^(∘)=32/sinB and 17/sin21^(∘)=c/sinC
We will solve the first equation for the measure of ∠B. First, let's rewrite the equation so that the sine of ∠B is isolated on the left-hand side.
Now we will use the inverse sine ratio.
sinB=32sin21^(∘)/17
⇕
m∠B=sin^(- 1)(32sin21^(∘)/17)
Let's approximate the value of m∠B to the nearest tenth of a degree using a calculator.
m∠B=sin^(- 1)(32sin21^(∘)/17)≈ 42.4^(∘)
The measure of ∠B is about 42.4^(∘). We will find the measure of the third angle ∠C using the Triangle Sum Theorem.
m∠C≈ 180^(∘)- 21^(∘)- 42.4^(∘)=116.6^(∘)
The measure of ∠C is about 116.6^(∘). Finally we will use the second equation to estimate the length c of the remaining side.
17/sin21^(∘)≈c/sin 116.6^(∘) [0.7em]
⇕ [0.7em]
c≈17sin116.6^(∘)/sin21^(∘)
Let's use a calculator.
c≈17sin116.6^(∘)/sin21^(∘)≈ 42.4
The length c of the remaining side of the triangle is about 42.4 units.
Solution II
In the second case we assume that ∠B is an obtuse angle.
Remember that the inverse sine function gives only acute angle measures. Therefore, in this case we will find m∠B using supplementary angles. In the first solution we have found the measure of the supplement of the current ∠B, about 42.4^(∘).
Now we can find the measures of ∠B and ∠C.
& m∠B≈ 180^(∘)- 42.4^(∘)=137.6^(∘)
& m∠C≈ 180^(∘)- 21^(∘)-137.6^(∘)=21.4^(∘)
Finally, let's find the value of c using the Law of Sines.
17/sin 21^(∘)≈c/sin21.4^(∘) ⇔ c≈ 17.3
The length c of the remaining side of the triangle is about 17.3 units.
