Exercise 29 Page 963

For any △ ABC, the Law of Cosines relates the cosine of each angle to the side lengths of the triangle.

To find the desired angle measure, we will start by drawing a diagram to illustrate the situation.

To find the value of m ∠ U, it will be useful to first find the value of s. We know that the lengths of t and u are 8 and 10 inches, respectively. We also know that their included angle has a measure of 96^(∘). With this information and using the Law of Cosines, we can write an equation to find s.

s^2= 10^2+ 8^2-2( 10)( 8)cos 96^(∘)

▼

Solve for s

CalcPow

Calculate power

s^2=100+64-2(10)(8)cos 96^(∘)

AddTerms

Add terms

s^2=164-2(10)(8)cos 96^(∘)

UseCalc

Use a calculator

s^2=164-160(-0.10452...)

MultNegNegOnePar

- a(- b)=a* b

s^2=164+16.72455...

AddTerms

Add terms

s^2=180.72455...

SqrtEqn

sqrt(LHS)=sqrt(RHS)

s=13.44338...

RoundDec

Round to 2 decimal place(s)

s≈ 13.44

Note that we only kept the principal root when solving the equation, because s is the length of a side. Let's add the value of s to our diagram.

Now, notice that we know the values of m ∠ S, as well as of s and u. This means that we use the Law of Sines to find the value of m ∠ U. Using this law, we can form the following equation. sin( m ∠ U)/u = sin( m ∠ S)/s Let's substitute the known values into the above equation and solve it for m∠ U.

sin( m∠ U)/u = sin( m ∠ S)/s

SubstituteValues

Substitute values

sin( m∠ U)/10 ≈ sin( 96)/13.44

▼

Solve for m∠ U

MultEqn

LHS * 10=RHS* 10

sin( m ∠ U) ≈ 10 sin(96)/13.44

UseCalc

Use a calculator

sin( m ∠ U) ≈ 0.73997...

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

m ∠ U ≈ sin^(-1) (0.73997...)

UseCalc

Use a calculator

m ∠ U ≈ 47.72900...

RoundDec

Round to 1 decimal place(s)

m ∠ U ≈ 47.7

Therefore, the value of m ∠ U is about 47.7^(∘).