Exercise 19 Page 924

First, let's draw the measurements from the exercise on a right triangle to visualize the given information.

We will find the missing measures one at a time. In this case, this means that we want to find m ∠ A, m ∠ B, and a.

Angle Measures

The cosine of ∠ A is the ratio of the length of the leg adjacent ∠ A to the length of the hypotenuse. cos A=Adjacent/Hypotenuse ⇒ cos A =5/10 By the definition of the inverse cosine, the inverse cosine of 510 is the measure of ∠ A. To find it, we can use a calculator.

m∠ A=cos ^(-1) 5/10

UseCalc

Use a calculator

m∠ A = 60.0 ^(∘)

To find m∠ B, recall that the acute angles of a right triangle are complementary. Therefore, m ∠ B and m ∠ A add up to 90^(∘). m ∠ B + m ∠ A = 90^(∘) Now, we can substitute the measure of ∠ A in our equation and find the measure of ∠ B. m ∠ B + 60.0^(∘) = 90^(∘) ⇔ m ∠ B =30.0^(∘)

Side Lengths

Finally, we can find the measure of a. To do it, we can use the Pythagorean Theorem. a^2 + b^2 = c^2 Let's substitute the known lengths, b = 5 and c= 10, into this equation to find a.

a^2+b^2=c^2

SubstituteII

b= 5, c= 10

a^2 + 5^2= 10^2

▼

Solve for a

CalcPow

Calculate power

a^2+25=100

SubEqn

LHS-25=RHS-25

a^2=75

SqrtEqn

sqrt(LHS)=sqrt(RHS)

a= sqrt(75)

UseCalc

Use a calculator

a = 8.66025 ...

RoundDec

Round to 1 decimal place(s)

a≈ 8.7